I've found that there is a bijection between integer partitions and ordered rooted trees with roots of degree 2 or greater. The rigorous proof is complicated, but the gist of it is that you take the Young diagram of a permutation and enclose it in a right isosceles triangle so that the hypotenuse of the triangle just touches the 'jagged' edge of the diagram, then collapse the negative space in the triangle along the hypotenuse. Negative 'corners' in the Young diagram correspond to leaves in the tree, and their depths correspond to the distance from the hypotenuse.
For example -
0 / \ 0 0 / \ 0 0
  
I've also noticed that adding a block to the Young diagram, which necessarily takes place at a negative corner, is equivalent to taking a tree's leaf and moving it one step closer to the root:
0  /|\  0 0 0  / 0
Thus, given a set of trees with the same number of vertices at each depth, it is necessarily the case that their corresponding Young diagrams have the same number of boxes in them - which is to say, their corresponding partitions are of the same integer. However, it is not the case that all partitions of a particular integer correspond to trees with the same number of vertices at each depth, or even trees with the same number of vertices overall, and likewise trees with the same total number of vertices can correspond to partitions of different integers.
This observation lead me to the question: Is there any way to tell, given a partition, how many vertices there are in the corresponding tree without doing something equivalent to finding the tree itself? And, is this bijection significant of a deeper underlying connection between partitions and trees, or just a coincidence?