Consider the problem of growing a random tree on a $L\times L$ square lattice of initially disconnected vertices, starting from an isolated vertex on one of the corners of the lattice and proceeding until reaching a Manhattan distance $L/2$ from it, according to the following rules:
At each step $n=0,\dots,L/2-1$, visit all the vertices at Manhattan distance $n$ from the origin. If the current vertex is connected to the tree, then with probability $p$ connect to it its two neighboring vertices at Manhattan distance $n+1$ from the origin, and with probability $1-p$ connect only one of the two. Skip adding edges that create a cycle (i.e., an edge to a neighbor that has already been added to the tree).
Question: how does the order of the tree grown by this process scale with $L$ depending on the choice of $p$?
It is clear that limiting cases are $O(L)$ and $O(L^2)$, but I am wondering whether there is also some sort of intermediate "fractal" dimension.