Easy to understand top-down DP solution

  • 9

    For a valid BST, there must be a node that is the root. The root can be any number between 1 and n. What if the root is k where k is in between 1 and n? It becomes obvious that 1 to k-1 should be k's left children and k+1 to n should be k's right children. Thus, a recursive formula is formulated as follows:

    number of BSTs with k being the root 
    = count of BSTs of k-1 consecutive numbers * count of BSTs of n-k consecutive numbers

    And since we have a total of n choices for k, total number of BSTs is thus a sum of all n choices for k.

    Code in Java:

    public int numTrees(int n) {
        return numTrees(n, new int[n]);
    private int numTrees(int n, int[] arr) {
        if(n<=1) return 1;
        if(arr[n-1] > 0) return arr[n-1];
        int num = 0;
        for(int i=1; i<=n; i++) 
            num += numTrees(i-1, arr) * numTrees(n-i, arr);
        arr[n-1] = num; // store for reuse
        return num;

  • 0

    What does this condition means ?

    if(arr[n-1] > 0) return arr[n-1];

  • 0

    arr is the array storing previously computed results. That is, arr[3] is the number of trees when n=3. Therefore,

    if(arr[n-1] > 0) return arr[n-1];

    means we simply return the stored result if it is already computed.

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