数据结构——二叉树的直径

原文地址：Diameter of a Binary Tree

树的直径（有时也称作宽度），指的是树中的两个叶子节点之间最长路径的节点的数目，下面的图显示了两个树的直径都是9，形成最长路径的两个端点的叶子节点被加上了阴影（注意每个树中长度为9的路径不止一个，但是没有比9再更长的路径了）。

一个树T的直径是下面几条里面最长的： - T的左子树的直径 - T的右子树的直径 - 两个叶子之间穿过T的根的最长路径（这个可以通过计算这个树的子树的高度获得）实现：

// Recursive optimized Java program to find the diameter of a// Binary Tree/* Class containing left and right child of current node and key value*/class Node{    int data;    Node left, right;    public Node(int item)    {        data = item;        left = right = null;    }}/* Class to print the Diameter */class BinaryTree{    Node root;    /* Method to calculate the diameter and return it to main */    int diameter(Node root)    {        /* base case if tree is empty */        if (root == null)            return 0;        /* get the height of left and right sub trees */        int lheight = height(root.left);        int rheight = height(root.right);        /* get the diameter of left and right subtrees */        int ldiameter = diameter(root.left);        int rdiameter = diameter(root.right);        /* Return max of following three          1) Diameter of left subtree         2) Diameter of right subtree         3) Height of left subtree + height of right subtree + 1 */        return Math.max(lheight + rheight + 1,                        Math.max(ldiameter, rdiameter));    }    /* A wrapper over diameter(Node root) */    int diameter()    {        return diameter(root);    }    /*The function Compute the "height" of a tree. Height is the      number f nodes along the longest path from the root node      down to the farthest leaf node.*/    static int height(Node node)    {        /* base case tree is empty */        if (node == null)            return 0;        /* If tree is not empty then height = 1 + max of left           height and right heights */        return (1 + Math.max(height(node.left), height(node.right)));    }    public static void main(String args[])    {        /* creating a binary tree and entering the nodes */        BinaryTree tree = new BinaryTree();        tree.root = new Node(1);        tree.root.left = new Node(2);        tree.root.right = new Node(3);        tree.root.left.left = new Node(4);        tree.root.left.right = new Node(5);        System.out.println("The diameter of given binary tree is : "                           + tree.diameter());    }}

时间复杂度：O(n^2)

输出：

Diameter of the given binary tree is 4

实现优化：以上实现可以通过在相同的递归，而不是分别调用height()来计算高度实现优化，感谢Amar 建议这个优化的版本。这个优化将时间减少到：O(n)。

// Recursive Java program to find the diameter of a// Binary Tree/* Class containing left and right child of current node and key value*/class Node{    int data;    Node left, right;    public Node(int item)    {        data = item;        left = right = null;    }}// A utility class to pass heigh objectclass Height{    int h;}/* Class to print the Diameter */class BinaryTree{    Node root;    /* define height =0 globally and  call diameterOpt(root,height)       from main */    int diameterOpt(Node root, Height height)    {        /* lh --> Height of left subtree           rh --> Height of right subtree */        Height lh = new Height(), rh = new Height();        if (root == null)        {            height.h = 0;            return 0; /* diameter is also 0 */        }        /* ldiameter  --> diameter of left subtree           rdiameter  --> Diameter of right subtree */         /* Get the heights of left and right subtrees in lh and rh         And store the returned values in ldiameter and ldiameter */        lh.h++;     rh.h++;         int ldiameter = diameterOpt(root.left, lh);        int rdiameter = diameterOpt(root.right, rh);        /* Height of current node is max of heights of left and         right subtrees plus 1*/        height.h = Math.max(lh.h, rh.h) + 1;        return Math.max(lh.h + rh.h + 1, Math.max(ldiameter, rdiameter));    }    /* A wrapper over diameter(Node root) */    int diameter()    {        Height height = new Height();        return diameterOpt(root, height);    }    /*The function Compute the "height" of a tree. Height is the      number f nodes along the longest path from the root node      down to the farthest leaf node.*/    static int height(Node node)    {        /* base case tree is empty */        if (node == null)            return 0;        /* If tree is not empty then height = 1 + max of left           height and right heights */        return (1 + Math.max(height(node.left), height(node.right)));    }    public static void main(String args[])    {        /* creating a binary tree and entering the nodes */        BinaryTree tree = new BinaryTree();        tree.root = new Node(1);        tree.root.left = new Node(2);        tree.root.right = new Node(3);        tree.root.left.left = new Node(4);        tree.root.left.right = new Node(5);        System.out.println("The diameter of given binary tree is : "                           + tree.diameter());    }}

时间复杂度：O(n)

输出：

4

参考文献：

http://www.cs.duke.edu/courses/spring00/cps100/assign/trees/diameter.html