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Problem solution · Go

Codeforces 9D — How many trees?

Codeforces 9D — How many trees?: a Go solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to EndlessCheng Codeforces Go.

Technique
Dynamic programming
Source
EndlessCheng Codeforces Go
Length
32 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Dynamic programming

For Codeforces 9D — How many trees?, the implementation records answers for smaller states and reuses them to build the requested result without repeating work.

  1. Define precisely what one DP state represents.
  2. Establish the base cases before transitions are evaluated.
  3. Process states in dependency order and combine only already-known values.

Code notes

  • 32 lines of Go from the credited upstream file 9D.go.
  • The implementation visibly relies on sequence storage, cached states.
  • No explicit loop blocks detected.

Complexity

Multiply the number of reachable states by the work performed for each transition, then include the stored state table in memory usage.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from EndlessCheng Codeforces Go by Σndless (EndlessCheng) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeCodeforces 9D — How many trees? · GoGo
Use this to learn the idea, then write your own version.
package main import (	. "fmt"	"io") // github.com/EndlessCheng/codeforces-gofunc CF9D(in io.Reader, out io.Writer) {	var N, H int	Fscan(in, &N, &H) 	// dp[n][h] 表示表示由 n 个点组成的高度不超过 h 的二叉树的个数	dp := make([][]int64, N+1)	for i := range dp {		dp[i] = make([]int64, N+1)	}	for i := range dp[0] {		dp[0][i] = 1	}	for h := 1; h <= N; h++ {		for n := 1; n <= N; n++ {			for left := 0; left < n; left++ {				dp[n][h] += dp[left][h-1] * dp[n-1-left][h-1]			}		}	}	Fprint(out, dp[N][N]-dp[N][H-1])} //func main() { CF9D(os.Stdin, os.Stdout) }

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