I was wondering whether there is a name for trees $\tau$ which could also represent coalesence as well as fragmentation in the following sense. Either a node $v$ in $\tau$ has child nodes (possibly $0$) which are not child nodes of any other node (as in a normal tree), or $v$ has exactly one child node $u$ which is also the only child node of another vertex $w$ where $v$ and $w$ have identical distance from the root node.
The motivation for this comes from soft matter physics where two bubbles may fuse when they come "too close" or one big bubble may disintegrate into multiple ones due to instability or even disappear if it is two small.
One can establish that $\tau$ is a planar graph but aside from that I am not quite sure what can be said and, therefore, I was looking for literature but I am lacking a name.