ISL 2005 C2

This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem:

Let $k$ be a nonnegative integer.

A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex $v$ is called an extended successor of a vertex $u$ if there is a chain of vertices $u_{0}=u$ , $u_{1}$ , $u_{2}$ , ..., $u_{t-1}$ , $u_{t}=v$ with $t>0$ such that the vertex $u_{i+1}$ is a successor of the vertex $u_{i}$ for every integer $i$ with $0\leq i\leq t-1$ . A vertex is called dynastic if it has two successors and each of these successors has at least $k$ extended successors.

Prove that if the forest has $n$ vertices, then there are at most $\frac{n}{k+2}$ dynastic vertices.