Losing Leaves

Here at the Benelux Advanced Phone Company (BAPC), we own the largest telephone network in the Benelux area. Unfortunately, the network has grown too large to manage properly, so BAPC has decided to sell part of it.

The network consists of interconnected transmission nodes. One node is the central switch. Every other node has exactly one upstream node. Following the upstream connections from any node always leads to the central switch. A node with no downstream nodes is a leaf, called a customer transmission node.

When selling, integrity must be kept. If BAPC sells a node $X$, it must also sell every node downstream of $X$.

BAPC will sell exactly $k$ transmission nodes. There may be many ways to choose them. BAPC wants to make its life as easy as possible, so it wants to minimize the number of customer transmission nodes left in the network after the sale, while keeping integrity.

For example, consider the second input case, shown in Figure L.1. The three striped red nodes are sold, and the two bold green nodes are the remaining customer nodes.

Figure L.1: Visualization of the second input case.

The input consists of:

• One line with two integers $n$ and $k$ ($0\leq k < n\leq 10^6$), the number of transmission nodes and the number of nodes to sell.
• $n-1$ lines, the $i$th of which contains one integer $p_i$ ($0 \leq p_i < i$), indicating that transmission node $i$ ($1 \leq i < n$) has an outgoing connection to node $p_i$.

The transmission nodes are numbered from $0$ to $n - 1$, inclusive. Node $0$ is always the central switch.

Output the minimum number of customer transmission nodes after selling $k$ transmission nodes. If the central switch is the only remaining node, it also counts as a customer node.