B 树

原文： https://www.programiz.com/dsa/b-plus-tree

在本教程中，您将学习什么是 B+ 树。 此外，您还将找到在 C，C++ ，Java 和 Python 中对 B+ 树进行搜索操作的工作示例。

B+ 树是自平衡树的高级形式，其中所有值都存在于叶级别中。

学习 B+ 树之前要理解的重要概念是多级索引。 在多级索引中，索引索引如下图所示创建。 它使访问数据更容易，更快捷。

使用 B+ 树进行多级索引

B+ 树的属性

1. 所有叶子处于同一级别。
2. 根至少有两个子级。
3. 除根节点外，每个节点最多可以包含m个子级，并且至少具有m / 2个子级。
4. 每个节点最多可以包含m -1键，以及最小⌈m/2⌉ -1键。

B 树和 B+ 树之间的比较

B 树

B+ 树

数据指针仅出现在 B+ 树的叶子节点上，而数据指针则出现在 B 树的内部，叶子或根节点上。

叶子在 B 树上不相互连接，而在 B+ 树上连接。

B+ 树上的操作比 B 树上的操作快。

B+ 树上的搜索

遵循以下步骤以在顺序为m的 B+ 树中搜索数据。 令要搜索的数据为k。

1. 从根节点开始。 比较k与根节点[k1, k2, k3, ... k(m-1)]上的键。
2. 如果k < k1，请转到根节点的左子节点。
3. 否则，如果k == k1，则比较k2。 如果k < k2，则k位于k1和k2之间。 因此，搜索k2的左子级。
4. 如果k > k2，则按照步骤 2 和步骤 6 中的k3，k4，... k(m-1)进行操作。
5. 重复上述步骤，直到到达叶节点。
6. 如果k在叶节点中存在，则返回true，否则返回false。

B+ 树上的搜索示例

让我们在以下 B+ 树上搜索k = 45。

B+ 树

1. 将k与根节点进行比较。
   
   在根下找不到

2. 由于k > 25，请找到合适的子级。
   
   移至根的右侧

3. 比较k与 35。由于k > 30，将k与 45 比较。
   
   未找到k

4. 由于k≥45，因此请找到合适的子级。
   
   移至右侧

5. 找到了k。
   

Python，Java 和 C/C++ 示例

 复制代码
# B+ tee in python
import math
# Node creation
class Node:
    def __init__(self, order):
        self.order = order
        self.values = []
        self.keys = []
        self.nextKey = None
        self.parent = None
        self.check_leaf = False
    # Insert at the leaf
    def insert_at_leaf(self, leaf, value, key):
        if (self.values):
            temp1 = self.values
            for i in range(len(temp1)):
                if (value == temp1[i]):
                    self.keys[i].append(key)
                    break
                elif (value < temp1[i]):
                    self.values = self.values[:i] + [value] + self.values[i:]
                    self.keys = self.keys[:i] + [[key]] + self.keys[i:]
                    break
                elif (i + 1 == len(temp1)):
                    self.values.append(value)
                    self.keys.append([key])
                    break
        else:
            self.values = [value]
            self.keys = [[key]]
# B plus tree
class BplusTree:
    def __init__(self, order):
        self.root = Node(order)
        self.root.check_leaf = True
    # Insert operation
    def insert(self, value, key):
        value = str(value)
        old_node = self.search(value)
        old_node.insert_at_leaf(old_node, value, key)
        if (len(old_node.values) == old_node.order):
            node1 = Node(old_node.order)
            node1.check_leaf = True
            node1.parent = old_node.parent
            mid = int(math.ceil(old_node.order / 2)) - 1
            node1.values = old_node.values[mid + 1:]
            node1.keys = old_node.keys[mid + 1:]
            node1.nextKey = old_node.nextKey
            old_node.values = old_node.values[:mid + 1]
            old_node.keys = old_node.keys[:mid + 1]
            old_node.nextKey = node1
            self.insert_in_parent(old_node, node1.values[0], node1)
    # Search operation for different operations
    def search(self, value):
        current_node = self.root
        while(current_node.check_leaf == False):
            temp2 = current_node.values
            for i in range(len(temp2)):
                if (value == temp2[i]):
                    current_node = current_node.keys[i + 1]
                    break
                elif (value < temp2[i]):
                    current_node = current_node.keys[i]
                    break
                elif (i + 1 == len(current_node.values)):
                    current_node = current_node.keys[i + 1]
                    break
        return current_node
    # Find the node
    def find(self, value, key):
        l = self.search(value)
        for i, item in enumerate(l.values):
            if item == value:
                if key in l.keys[i]:
                    return True
                else:
                    return False
        return False
    # Inserting at the parent
    def insert_in_parent(self, n, value, ndash):
        if (self.root == n):
            rootNode = Node(n.order)
            rootNode.values = [value]
            rootNode.keys = [n, ndash]
            self.root = rootNode
            n.parent = rootNode
            ndash.parent = rootNode
            return
        parentNode = n.parent
        temp3 = parentNode.keys
        for i in range(len(temp3)):
            if (temp3[i] == n):
                parentNode.values = parentNode.values[:i] + \
                    [value] + parentNode.values[i:]
                parentNode.keys = parentNode.keys[:i +
                                                  1] + [ndash] + parentNode.keys[i + 1:]
                if (len(parentNode.keys) > parentNode.order):
                    parentdash = Node(parentNode.order)
                    parentdash.parent = parentNode.parent
                    mid = int(math.ceil(parentNode.order / 2)) - 1
                    parentdash.values = parentNode.values[mid + 1:]
                    parentdash.keys = parentNode.keys[mid + 1:]
                    value_ = parentNode.values[mid]
                    if (mid == 0):
                        parentNode.values = parentNode.values[:mid + 1]
                    else:
                        parentNode.values = parentNode.values[:mid]
                    parentNode.keys = parentNode.keys[:mid + 1]
                    for j in parentNode.keys:
                        j.parent = parentNode
                    for j in parentdash.keys:
                        j.parent = parentdash
                    self.insert_in_parent(parentNode, value_, parentdash)
    # Delete a node
    def delete(self, value, key):
        node_ = self.search(value)
        temp = 0
        for i, item in enumerate(node_.values):
            if item == value:
                temp = 1
                if key in node_.keys[i]:
                    if len(node_.keys[i]) > 1:
                        node_.keys[i].pop(node_.keys[i].index(key))
                    elif node_ == self.root:
                        node_.values.pop(i)
                        node_.keys.pop(i)
                    else:
                        node_.keys[i].pop(node_.keys[i].index(key))
                        del node_.keys[i]
                        node_.values.pop(node_.values.index(value))
                        self.deleteEntry(node_, value, key)
                else:
                    print("Value not in Key")
                    return
        if temp == 0:
            print("Value not in Tree")
            return
    # Delete an entry
    def deleteEntry(self, node_, value, key):
        if not node_.check_leaf:
            for i, item in enumerate(node_.keys):
                if item == key:
                    node_.keys.pop(i)
                    break
            for i, item in enumerate(node_.values):
                if item == value:
                    node_.values.pop(i)
                    break
        if self.root == node_ and len(node_.keys) == 1:
            self.root = node_.keys[0]
            node_.keys[0].parent = None
            del node_
            return
        elif (len(node_.keys) < int(math.ceil(node_.order / 2)) and node_.check_leaf == False) or (len(node_.values) < int(math.ceil((node_.order - 1) / 2)) and node_.check_leaf == True):
            is_predecessor = 0
            parentNode = node_.parent
            PrevNode = -1
            NextNode = -1
            PrevK = -1
            PostK = -1
            for i, item in enumerate(parentNode.keys):
                if item == node_:
                    if i > 0:
                        PrevNode = parentNode.keys[i - 1]
                        PrevK = parentNode.values[i - 1]
                    if i < len(parentNode.keys) - 1:
                        NextNode = parentNode.keys[i + 1]
                        PostK = parentNode.values[i]
            if PrevNode == -1:
                ndash = NextNode
                value_ = PostK
            elif NextNode == -1:
                is_predecessor = 1
                ndash = PrevNode
                value_ = PrevK
            else:
                if len(node_.values) + len(NextNode.values) < node_.order:
                    ndash = NextNode
                    value_ = PostK
                else:
                    is_predecessor = 1
                    ndash = PrevNode
                    value_ = PrevK
            if len(node_.values) + len(ndash.values) < node_.order:
                if is_predecessor == 0:
                    node_, ndash = ndash, node_
                ndash.keys += node_.keys
                if not node_.check_leaf:
                    ndash.values.append(value_)
                else:
                    ndash.nextKey = node_.nextKey
                ndash.values += node_.values
                if not ndash.check_leaf:
                    for j in ndash.keys:
                        j.parent = ndash
                self.deleteEntry(node_.parent, value_, node_)
                del node_
            else:
                if is_predecessor == 1:
                    if not node_.check_leaf:
                        ndashpm = ndash.keys.pop(-1)
                        ndashkm_1 = ndash.values.pop(-1)
                        node_.keys = [ndashpm] + node_.keys
                        node_.values = [value_] + node_.values
                        parentNode = node_.parent
                        for i, item in enumerate(parentNode.values):
                            if item == value_:
                                p.values[i] = ndashkm_1
                                break
                    else:
                        ndashpm = ndash.keys.pop(-1)
                        ndashkm = ndash.values.pop(-1)
                        node_.keys = [ndashpm] + node_.keys
                        node_.values = [ndashkm] + node_.values
                        parentNode = node_.parent
                        for i, item in enumerate(p.values):
                            if item == value_:
                                parentNode.values[i] = ndashkm
                                break
                else:
                    if not node_.check_leaf:
                        ndashp0 = ndash.keys.pop(0)
                        ndashk0 = ndash.values.pop(0)
                        node_.keys = node_.keys + [ndashp0]
                        node_.values = node_.values + [value_]
                        parentNode = node_.parent
                        for i, item in enumerate(parentNode.values):
                            if item == value_:
                                parentNode.values[i] = ndashk0
                                break
                    else:
                        ndashp0 = ndash.keys.pop(0)
                        ndashk0 = ndash.values.pop(0)
                        node_.keys = node_.keys + [ndashp0]
                        node_.values = node_.values + [ndashk0]
                        parentNode = node_.parent
                        for i, item in enumerate(parentNode.values):
                            if item == value_:
                                parentNode.values[i] = ndash.values[0]
                                break
                if not ndash.check_leaf:
                    for j in ndash.keys:
                        j.parent = ndash
                if not node_.check_leaf:
                    for j in node_.keys:
                        j.parent = node_
                if not parentNode.check_leaf:
                    for j in parentNode.keys:
                        j.parent = parentNode
# Print the tree
def printTree(tree):
    lst = [tree.root]
    level = [0]
    leaf = None
    flag = 0
    lev_leaf = 0
    node1 = Node(str(level[0]) + str(tree.root.values))
    while (len(lst) != 0):
        x = lst.pop(0)
        lev = level.pop(0)
        if (x.check_leaf == False):
            for i, item in enumerate(x.keys):
                print(item.values)
        else:
            for i, item in enumerate(x.keys):
                print(item.values)
            if (flag == 0):
                lev_leaf = lev
                leaf = x
                flag = 1
record_len = 3
bplustree = BplusTree(record_len)
bplustree.insert('5', '33')
bplustree.insert('15', '21')
bplustree.insert('25', '31')
bplustree.insert('35', '41')
bplustree.insert('45', '10')
printTree(bplustree)
if(bplustree.find('5', '34')):
    print("Found")
else:
    print("Not found")

 复制代码
// Searching on a B+ tree in Java
import java.util.*;
public class BPlusTree {
  int m;
  InternalNode root;
  LeafNode firstLeaf;
  // Binary search program
  private int binarySearch(DictionaryPair[] dps, int numPairs, int t) {
    Comparator<DictionaryPair> c = new Comparator<DictionaryPair>() {
      @Override
      public int compare(DictionaryPair o1, DictionaryPair o2) {
        Integer a = Integer.valueOf(o1.key);
        Integer b = Integer.valueOf(o2.key);
        return a.compareTo(b);
      }
    };
    return Arrays.binarySearch(dps, 0, numPairs, new DictionaryPair(t, 0), c);
  }
  // Find the leaf node
  private LeafNode findLeafNode(int key) {
    Integer[] keys = this.root.keys;
    int i;
    for (i = 0; i < this.root.degree - 1; i++) {
      if (key < keys[i]) {
        break;
      }
    }
    Node child = this.root.childPointers[i];
    if (child instanceof LeafNode) {
      return (LeafNode) child;
    } else {
      return findLeafNode((InternalNode) child, key);
    }
  }
  // Find the leaf node
  private LeafNode findLeafNode(InternalNode node, int key) {
    Integer[] keys = node.keys;
    int i;
    for (i = 0; i < node.degree - 1; i++) {
      if (key < keys[i]) {
        break;
      }
    }
    Node childNode = node.childPointers[i];
    if (childNode instanceof LeafNode) {
      return (LeafNode) childNode;
    } else {
      return findLeafNode((InternalNode) node.childPointers[i], key);
    }
  }
  // Finding the index of the pointer
  private int findIndexOfPointer(Node[] pointers, LeafNode node) {
    int i;
    for (i = 0; i < pointers.length; i++) {
      if (pointers[i] == node) {
        break;
      }
    }
    return i;
  }
  // Get the mid point
  private int getMidpoint() {
    return (int) Math.ceil((this.m + 1) / 2.0) - 1;
  }
  // Balance the tree
  private void handleDeficiency(InternalNode in) {
    InternalNode sibling;
    InternalNode parent = in.parent;
    if (this.root == in) {
      for (int i = 0; i < in.childPointers.length; i++) {
        if (in.childPointers[i] != null) {
          if (in.childPointers[i] instanceof InternalNode) {
            this.root = (InternalNode) in.childPointers[i];
            this.root.parent = null;
          } else if (in.childPointers[i] instanceof LeafNode) {
            this.root = null;
          }
        }
      }
    }
    else if (in.leftSibling != null && in.leftSibling.isLendable()) {
      sibling = in.leftSibling;
    } else if (in.rightSibling != null && in.rightSibling.isLendable()) {
      sibling = in.rightSibling;
      int borrowedKey = sibling.keys[0];
      Node pointer = sibling.childPointers[0];
      in.keys[in.degree - 1] = parent.keys[0];
      in.childPointers[in.degree] = pointer;
      parent.keys[0] = borrowedKey;
      sibling.removePointer(0);
      Arrays.sort(sibling.keys);
      sibling.removePointer(0);
      shiftDown(in.childPointers, 1);
    } else if (in.leftSibling != null && in.leftSibling.isMergeable()) {
    } else if (in.rightSibling != null && in.rightSibling.isMergeable()) {
      sibling = in.rightSibling;
      sibling.keys[sibling.degree - 1] = parent.keys[parent.degree - 2];
      Arrays.sort(sibling.keys, 0, sibling.degree);
      parent.keys[parent.degree - 2] = null;
      for (int i = 0; i < in.childPointers.length; i++) {
        if (in.childPointers[i] != null) {
          sibling.prependChildPointer(in.childPointers[i]);
          in.childPointers[i].parent = sibling;
          in.removePointer(i);
        }
      }
      parent.removePointer(in);
      sibling.leftSibling = in.leftSibling;
    }
    if (parent != null && parent.isDeficient()) {
      handleDeficiency(parent);
    }
  }
  private boolean isEmpty() {
    return firstLeaf == null;
  }
  private int linearNullSearch(DictionaryPair[] dps) {
    for (int i = 0; i < dps.length; i++) {
      if (dps[i] == null) {
        return i;
      }
    }
    return -1;
  }
  private int linearNullSearch(Node[] pointers) {
    for (int i = 0; i < pointers.length; i++) {
      if (pointers[i] == null) {
        return i;
      }
    }
    return -1;
  }
  private void shiftDown(Node[] pointers, int amount) {
    Node[] newPointers = new Node[this.m + 1];
    for (int i = amount; i < pointers.length; i++) {
      newPointers[i - amount] = pointers[i];
    }
    pointers = newPointers;
  }
  private void sortDictionary(DictionaryPair[] dictionary) {
    Arrays.sort(dictionary, new Comparator<DictionaryPair>() {
      @Override
      public int compare(DictionaryPair o1, DictionaryPair o2) {
        if (o1 == null && o2 == null) {
          return 0;
        }
        if (o1 == null) {
          return 1;
        }
        if (o2 == null) {
          return -1;
        }
        return o1.compareTo(o2);
      }
    });
  }
  private Node[] splitChildPointers(InternalNode in, int split) {
    Node[] pointers = in.childPointers;
    Node[] halfPointers = new Node[this.m + 1];
    for (int i = split + 1; i < pointers.length; i++) {
      halfPointers[i - split - 1] = pointers[i];
      in.removePointer(i);
    }
    return halfPointers;
  }
  private DictionaryPair[] splitDictionary(LeafNode ln, int split) {
    DictionaryPair[] dictionary = ln.dictionary;
    DictionaryPair[] halfDict = new DictionaryPair[this.m];
    for (int i = split; i < dictionary.length; i++) {
      halfDict[i - split] = dictionary[i];
      ln.delete(i);
    }
    return halfDict;
  }
  private void splitInternalNode(InternalNode in) {
    InternalNode parent = in.parent;
    int midpoint = getMidpoint();
    int newParentKey = in.keys[midpoint];
    Integer[] halfKeys = splitKeys(in.keys, midpoint);
    Node[] halfPointers = splitChildPointers(in, midpoint);
    in.degree = linearNullSearch(in.childPointers);
    InternalNode sibling = new InternalNode(this.m, halfKeys, halfPointers);
    for (Node pointer : halfPointers) {
      if (pointer != null) {
        pointer.parent = sibling;
      }
    }
    sibling.rightSibling = in.rightSibling;
    if (sibling.rightSibling != null) {
      sibling.rightSibling.leftSibling = sibling;
    }
    in.rightSibling = sibling;
    sibling.leftSibling = in;
    if (parent == null) {
      Integer[] keys = new Integer[this.m];
      keys[0] = newParentKey;
      InternalNode newRoot = new InternalNode(this.m, keys);
      newRoot.appendChildPointer(in);
      newRoot.appendChildPointer(sibling);
      this.root = newRoot;
      in.parent = newRoot;
      sibling.parent = newRoot;
    } else {
      parent.keys[parent.degree - 1] = newParentKey;
      Arrays.sort(parent.keys, 0, parent.degree);
      int pointerIndex = parent.findIndexOfPointer(in) + 1;
      parent.insertChildPointer(sibling, pointerIndex);
      sibling.parent = parent;
    }
  }
  private Integer[] splitKeys(Integer[] keys, int split) {
    Integer[] halfKeys = new Integer[this.m];
    keys[split] = null;
    for (int i = split + 1; i < keys.length; i++) {
      halfKeys[i - split - 1] = keys[i];
      keys[i] = null;
    }
    return halfKeys;
  }
  public void insert(int key, double value) {
    if (isEmpty()) {
      LeafNode ln = new LeafNode(this.m, new DictionaryPair(key, value));
      this.firstLeaf = ln;
    } else {
      LeafNode ln = (this.root == null) ? this.firstLeaf : findLeafNode(key);
      if (!ln.insert(new DictionaryPair(key, value))) {
        ln.dictionary[ln.numPairs] = new DictionaryPair(key, value);
        ln.numPairs++;
        sortDictionary(ln.dictionary);
        int midpoint = getMidpoint();
        DictionaryPair[] halfDict = splitDictionary(ln, midpoint);
        if (ln.parent == null) {
          Integer[] parent_keys = new Integer[this.m];
          parent_keys[0] = halfDict[0].key;
          InternalNode parent = new InternalNode(this.m, parent_keys);
          ln.parent = parent;
          parent.appendChildPointer(ln);
        } else {
          int newParentKey = halfDict[0].key;
          ln.parent.keys[ln.parent.degree - 1] = newParentKey;
          Arrays.sort(ln.parent.keys, 0, ln.parent.degree);
        }
        LeafNode newLeafNode = new LeafNode(this.m, halfDict, ln.parent);
        int pointerIndex = ln.parent.findIndexOfPointer(ln) + 1;
        ln.parent.insertChildPointer(newLeafNode, pointerIndex);
        newLeafNode.rightSibling = ln.rightSibling;
        if (newLeafNode.rightSibling != null) {
          newLeafNode.rightSibling.leftSibling = newLeafNode;
        }
        ln.rightSibling = newLeafNode;
        newLeafNode.leftSibling = ln;
        if (this.root == null) {
          this.root = ln.parent;
        } else {
          InternalNode in = ln.parent;
          while (in != null) {
            if (in.isOverfull()) {
              splitInternalNode(in);
            } else {
              break;
            }
            in = in.parent;
          }
        }
      }
    }
  }
  public Double search(int key) {
    if (isEmpty()) {
      return null;
    }
    LeafNode ln = (this.root == null) ? this.firstLeaf : findLeafNode(key);
    DictionaryPair[] dps = ln.dictionary;
    int index = binarySearch(dps, ln.numPairs, key);
    if (index < 0) {
      return null;
    } else {
      return dps[index].value;
    }
  }
  public ArrayList<Double> search(int lowerBound, int upperBound) {
    ArrayList<Double> values = new ArrayList<Double>();
    LeafNode currNode = this.firstLeaf;
    while (currNode != null) {
      DictionaryPair dps[] = currNode.dictionary;
      for (DictionaryPair dp : dps) {
        if (dp == null) {
          break;
        }
        if (lowerBound <= dp.key && dp.key <= upperBound) {
          values.add(dp.value);
        }
      }
      currNode = currNode.rightSibling;
    }
    return values;
  }
  public BPlusTree(int m) {
    this.m = m;
    this.root = null;
  }
  public class Node {
    InternalNode parent;
  }
  private class InternalNode extends Node {
    int maxDegree;
    int minDegree;
    int degree;
    InternalNode leftSibling;
    InternalNode rightSibling;
    Integer[] keys;
    Node[] childPointers;
    private void appendChildPointer(Node pointer) {
      this.childPointers[degree] = pointer;
      this.degree++;
    }
    private int findIndexOfPointer(Node pointer) {
      for (int i = 0; i < childPointers.length; i++) {
        if (childPointers[i] == pointer) {
          return i;
        }
      }
      return -1;
    }
    private void insertChildPointer(Node pointer, int index) {
      for (int i = degree - 1; i >= index; i--) {
        childPointers[i + 1] = childPointers[i];
      }
      this.childPointers[index] = pointer;
      this.degree++;
    }
    private boolean isDeficient() {
      return this.degree < this.minDegree;
    }
    private boolean isLendable() {
      return this.degree > this.minDegree;
    }
    private boolean isMergeable() {
      return this.degree == this.minDegree;
    }
    private boolean isOverfull() {
      return this.degree == maxDegree + 1;
    }
    private void prependChildPointer(Node pointer) {
      for (int i = degree - 1; i >= 0; i--) {
        childPointers[i + 1] = childPointers[i];
      }
      this.childPointers[0] = pointer;
      this.degree++;
    }
    private void removeKey(int index) {
      this.keys[index] = null;
    }
    private void removePointer(int index) {
      this.childPointers[index] = null;
      this.degree--;
    }
    private void removePointer(Node pointer) {
      for (int i = 0; i < childPointers.length; i++) {
        if (childPointers[i] == pointer) {
          this.childPointers[i] = null;
        }
      }
      this.degree--;
    }
    private InternalNode(int m, Integer[] keys) {
      this.maxDegree = m;
      this.minDegree = (int) Math.ceil(m / 2.0);
      this.degree = 0;
      this.keys = keys;
      this.childPointers = new Node[this.maxDegree + 1];
    }
    private InternalNode(int m, Integer[] keys, Node[] pointers) {
      this.maxDegree = m;
      this.minDegree = (int) Math.ceil(m / 2.0);
      this.degree = linearNullSearch(pointers);
      this.keys = keys;
      this.childPointers = pointers;
    }
  }
  public class LeafNode extends Node {
    int maxNumPairs;
    int minNumPairs;
    int numPairs;
    LeafNode leftSibling;
    LeafNode rightSibling;
    DictionaryPair[] dictionary;
    public void delete(int index) {
      this.dictionary[index] = null;
      numPairs--;
    }
    public boolean insert(DictionaryPair dp) {
      if (this.isFull()) {
        return false;
      } else {
        this.dictionary[numPairs] = dp;
        numPairs++;
        Arrays.sort(this.dictionary, 0, numPairs);
        return true;
      }
    }
    public boolean isDeficient() {
      return numPairs < minNumPairs;
    }
    public boolean isFull() {
      return numPairs == maxNumPairs;
    }
    public boolean isLendable() {
      return numPairs > minNumPairs;
    }
    public boolean isMergeable() {
      return numPairs == minNumPairs;
    }
    public LeafNode(int m, DictionaryPair dp) {
      this.maxNumPairs = m - 1;
      this.minNumPairs = (int) (Math.ceil(m / 2) - 1);
      this.dictionary = new DictionaryPair[m];
      this.numPairs = 0;
      this.insert(dp);
    }
    public LeafNode(int m, DictionaryPair[] dps, InternalNode parent) {
      this.maxNumPairs = m - 1;
      this.minNumPairs = (int) (Math.ceil(m / 2) - 1);
      this.dictionary = dps;
      this.numPairs = linearNullSearch(dps);
      this.parent = parent;
    }
  }
  public class DictionaryPair implements Comparable<DictionaryPair> {
    int key;
    double value;
    public DictionaryPair(int key, double value) {
      this.key = key;
      this.value = value;
    }
    public int compareTo(DictionaryPair o) {
      if (key == o.key) {
        return 0;
      } else if (key > o.key) {
        return 1;
      } else {
        return -1;
      }
    }
  }
  public static void main(String[] args) {
    BPlusTree bpt = null;
    bpt = new BPlusTree(3);
    bpt.insert(5, 33);
    bpt.insert(15, 21);
    bpt.insert(25, 31);
    bpt.insert(35, 41);
    bpt.insert(45, 10);
    if (bpt.search(15) != null) {
      System.out.println("Found");
    } else {
      System.out.println("Not Found");
    }
    ;
  }
}