Forest

Consider a simple model of tree growth in a pine forest. All trees are divided into four groups: young seedlings (age group from $0$ to $15$), young trees ($16$ to $30$), middle-aged trees ($31$ to $45$), and old trees (older than $45$). The duration of one period is $15$ years. In this problem, you need to determine changes in the population of trees if:

1. A certain percentage of each group of trees dies;

2. The surviving trees move to the next age group. Old trees remain old;

3. Dead trees are replaced by young seedlings. Thus, the number of trees in the forest stays constant.

Initially, the forest consists of $n$ young seedlings. The percentages of trees that die in each group per period are known: p_{baby}, p_{young}, p_{middle-age}, p_{old}. You need to determine the average number of trees in each group after $15k$ years.