biximo's blog

Proving certain binary tree property
By biximo, history, 11 months ago, In English

I recently came across a claim about binary trees that I was unable to prove. Given a binary tree, $$$\sum{|child_l|\times |child_r| } = O(N^2)$$$

Could someone provide proof and/or a way to intuitively explain this?

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11 months ago, # |
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Actually, I guess I have some kind of intuitive explanation behind this. If we imagine that each node "contributes" to the $$$N^2$$$ count, we can observe that no matter how the tree is structured, the total number that it contributes will be maximum $$$N$$$, since once two subtrees "merge" they will no longer multiply within themselves to count towards the $$$N^2$$$. So, since each node will add a maximum of $$$N$$$ towards the count, the total number will add up to maximum $$$N^2$$$. I think I am more interested in formal proof.

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11 months ago, # |
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Another way to think about it is that any pair of nodes $$$(u, v)$$$ is considered at most (exactly, in fact) once and this happens at their LCA.

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11 months ago, # |
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Suppose we want to count number of pairs $$$(u, v)$$$ such that $$$LCA(u, v) = i$$$. If children of $$$i$$$ are $$$l$$$ and $$$r$$$, then the number is equal of $$$\text{SubtreeSize}(l) \times \text{SubtreeSize}(r)$$$ if we consider pairs $$$(u, v)$$$ and $$$(v, u)$$$ equal. Therefore the sum you mentioned is equal to total number of pairs of vertices in the tree (number of subsets of size two).

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    11 months ago, # ^ |
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    Total number of pairs except the ones $$$\{u,v\}$$$ where $$$u$$$ is an ancestor of $$$v$$$ since $$$LCA(u,v) = u$$$ but this is not counted in $$$\text{SubtreeSize} (l_u) \times \text{SubtreeSize} (r_u)$$$

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11 months ago, # |
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The given sum is same as the union of cross products of sets of vertoces of left and right subtrees. Now we can see that each unordered pair of vertices $$$(u,v)$$$ occurs atmost once in the above union. This also gives an upper bound of $$$N*(N-1)/2$$$

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11 months ago, # |
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https://mirror.codeforces.com/blog/entry/124538?#comment-1105664

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