Imagine growing trees from $k$ seeds on a square $n \times n$ region
of $\mathbb{Z}^2$.
At each step, a unit-length edge $e$ between two points of
$\mathbb{Z}^2$ is added.
The edge $e$ is chosen randomly among those that have one endpoint
touching an existing tree and the other endpoint not touching
any tree. Edge $e$ then extends the tree $T$ it touches, maintains
$T$ a tree, and does not join to another tree.
The process is repeated until no such edges remain.
Here is a view of (a portion of) the trees grown from three seeds:

I expected the trees to intertwine and tangle with one another
as they grew,
but instead they appear to be approximating the Voronoi
diagram of the seeds.
Here is an example (of which the above is a detail)
of $k=10$ trees grown within a $100 \times 100$ square:

I have overlaid the Voronoi diagram of the seeds.

My primary question is:

Q1. Is it known that a tree-growth process like mine approximates the Voronoi diagram in some sense?

Short of this, I would like to correct my faulty intuition:

Q2. Why do the trees not entangle more significantly where they meet?

Finally:

Q3. Are there any results known on the structure of the individual grown trees? For example, their expected height, perhaps as a function of their Voronoi region shape? (Not surprisingly, the tallest tree ($h=110$) above is the mauve one with root $(61,41)$ that encompasses the upper right corner.)

Although the book *Percolation* has a chapter on
"Random Voronoi Percolation,"
I have not been successful in matching their models to mine
closely enough to draw any definite conclusions.

Thanks for any pointers to relevant literature!