KöMaL 2018 B4928

The trunk of an ever-growing tree forks in two at a height of one foot. In the following, the term branch will refer to a section between two joints, with no further joint along its length. Every branch of the ever-growing tree is straight, and terminates one foot higher than its lower end. The branches starting from the upper end of the branch are considered the children of the branch, also called the siblings of each other. Every branch of the tree has at least two children. If a branch does not have exactly two children then it has a sibling with exactly two children. Siblings always have different numbers of children. If a branch has more than two children then it has a sibling with one fewer children. How many branches start from joints at a height of $n$ feet?