Cat in a tree

A cat lives in a tree that has $N$ nodes. She will demarcate her territory by “marking” some of the tree nodes. Marked nodes may not be closer to each other than distance $D$, or the smell will be too overwhelming. Find the maximum number of nodes that the cat can mark.

First line has two integers, $N$ and $D$. The $0$-th node is the root node of the tree. Then follows $N-1$ lines, the $i$-th of which contain a single integer $x_ i$ with $0 \leq x_ i < i$ (starting with $i = 1$). This means that node $x_ i$ is connected to node $i$.

We always have $1 \leq N, D \leq 2 \cdot 10^5$.

Output should contain one integer: the maximum number of nodes that can be marked.

Sample Input 1 | Sample Output 1 |
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4 3 0 0 1 |
2 |

Sample Input 2 | Sample Output 2 |
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3 1000 0 0 |
1 |