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完全二叉树节点数

Créé le : 28 janvier 2025

Répondu en utilisant O1 par Chat01

Question

有一棵完全二叉树，它的第10层有10个叶子节点，那么这棵树最多有多少个节点？

Réflexion

Examining the structure

I'm thinking through a complete binary tree, starting from level 1 and reaching level 10 where 10 leaf nodes appear. The question is about the maximum possible total number of nodes in this tree.

Mapping the structure

Piecing together, a 'complete binary tree' fills all levels except possibly the last, where nodes align leftward. The 10th level of the tree, featuring 10 leaf nodes, suggests a specific node distribution.

Evaluating tree depth

OK, let me see. If the 10th level is indeed the final leaf level, then adding extra levels would mean those nodes wouldn't be considered leaves.

Pinpointing leaf locations

In a complete binary tree, leaf nodes are typically found in the last or second-to-last level, depending on whether the last level is fully occupied.

Placing leaf nodes

In a complete binary tree, to maximize nodes, you'd fully fill levels 1 through 9. Then, the 10th level accommodates exactly 10 leftmost leaf nodes.

Clarifying the structure

I'm seeing that all 10 nodes at the 10th level are leaf nodes, with no extra nodes at level 11. This confirms the tree's structure with 10 leaves at level 10.

Assessing node capacity

In a complete binary tree, the first 9 levels can hold 511 nodes. Adding nodes to level 10, which already contains 10 leaf nodes, requires a thorough calculation to maximize the tree's capacity.

Maximizing level 10

I’m thinking through level 10's potential, which can hold 512 nodes. Since 10 nodes are already leaves, adding internal nodes would extend the tree to level 11, not altering the 10 leaves at level 10.

Evaluating level structure

It's worth noting that the question's ambiguity means we must carefully assess the possibility of additional non-leaf nodes at level 10, potentially extending the tree to level 11 or beyond.

Counting nodes strategically

OK, let's see. The key observation is that the 10th level's 10 leaf nodes might indicate a larger tree with potential deeper levels, each adding more nodes.

Maximizing node distribution

By populating the initial 9 levels completely, ensuring 10 leaves at level 10, and extending through levels 11 and 12 as needed, the tree's total nodes are optimized.

Maximizing nodes

Filling level 10 to capacity with 10 leaves means the tree extends to level 11, where non-leaf nodes produce at least 502 children, potentially adding many more nodes.

Ensuring completeness

To keep the tree complete, all levels up to level 10 must be filled. Thus, if any node at level 10 has children, level 11 will still be fully populated.

Balancing node distribution

I'm thinking through how to balance node distribution in a complete binary tree. Ensuring 10 leaf nodes at level 10 while maximizing node count in deeper levels is the challenge.

Mapping out node arrangement

I’m noting that the 10 leaf nodes must be placed at the right end of level 10, ensuring all leftward nodes are occupied before any children are added to the next level.

Strategizing node distribution

I’m mapping out the idea that first filling the 502 leftmost nodes at level 10 with two children each, then letting the next 10 nodes be leaves, could optimize the tree's final node count.

Piecing together node boundaries

OK, let's ensure 10 leaf nodes by placing a boundary so the last 10 nodes in level 10 are leaves, while the rest up to 512 have children, keeping the tree complete.

Allocating resources

Mapping out the distribution of nodes: exactly 10 leaf nodes at level 10 with the rest as internal nodes, and each internal node having either 1 or 2 children.

Mapping node structure

Filling level 11 from left to right, each internal node at level 10 can have 1 or 2 children, leading to a node count between I10 and 2*I10.

Maximizing node expansion

I’m figuring out how letting each node at level 10 have up to 2 children could push the tree to level 11 or 12, potentially maximizing the total node count.

Balancing tree growth

I'm exploring the implications of unlimited tree height while ensuring 10 leaves at level 10, considering that each node can yield infinite nodes in deeper levels.

Confirming completeness

OK, let me see: In a complete binary tree of height 10, wherein the 10th level consists uniquely of 10 leaf nodes, the rest of the tree must be fully filled.

Confirming total nodes

I’ve concluded that with levels 1 through 9 fully populated and 10 more nodes at level 10, the complete binary tree sums up to a total of 521 nodes.

Thought about 完全二叉树节点，持续 1m 3s

Réponse

题意可以理解为：“这是一棵完全二叉树，第 10 层恰好有 10 个叶子节点。在这种情况下，这棵树最多能有多少个节点？”

关键思路

  1. 完全二叉树的定义
    所谓“完全二叉树”，是指除了最末层可能不满之外，其它各层都完全填满，并且最末层的节点都尽可能地集中在左侧。

  2. 层数与节点数的关系

    • 如果一棵二叉树高度（层数）是 hh，并且前 h−1h-1 层都是满的，那么前 h−1h-1 层的节点总数为 2h−1−1 2^{h-1} - 1
    • 第 hh 层最多可以容纳 2h−12^{h-1} 个节点（如果这一层是满的情况）。

  3. 题目所给的条件

    • “第 10 层有 10 个叶子节点”通常可理解为：第 10 层是该完全二叉树的最末层，里面一共出现了 10 个节点，且它们都是叶子（没有更深的层次）。
    • 在完全二叉树中，只有最后一层可以不满，但上一层及更上层一定是满的。因此，第 1～9 层必须全部填满。

  4. 计算

    • 前 9 层为满二叉树： 前 9 层节点数=29−1=512−1=511 \text{前 9 层节点数} = 2^9 - 1 = 512 - 1 = 511
    • 第 10 层有 10 个（且都是叶子）： 第 10 层节点数=10 \text{第 10 层节点数} = 10
    • 总节点数： 511+10=521 511 + 10 = 521

因此，这棵完全二叉树在“第 10 层正好有 10 个叶子”的前提下，最多拥有 521521 个节点。

常见疑问补充

  • 为什么不继续添加第 11 层？
    若存在第 11 层，那么第 10 层就不再是“全部是叶子”的层；因为有节点在第 10 层会拥有子节点（导致它们不是叶子），和题目“第 10 层有 10 个叶子”这一描述相冲突。通常这类题目默认“有叶子出现的那一层就是最末层”。

  • 是否可能第 10 层除了这 10 个节点外还有其他节点？
    若还要再添加更多节点在第 10 层，则“第 10 层有 10 个叶子”就不成立了（会变成“第 10 层有大于 10 个节点”）。题意中一般理解为：最末层出现的节点总计就是 10 个，并且它们都是叶子。

结论：在给定条件下，这棵完全二叉树最多有 521 个节点。

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