Here are some pictures of the random tree reduced over a finite number of nodes.

Over $32$ nodes:
![The random tree over 32 nodes.][1]

Over $96$ nodes:
![The random tree over 96 nodes.][2]

Over $512$ nodes:
![The random tree over 512 nodes.][3]

I employ the following ${\tt Sage}$ code to generate these:

    def random_tree(n) :
    tree = Graph()
    tree.add_vertex(1)
    for i in xrange(2, n + 1) :
        tree.add_edge(i, randint(1, i - 1))
    return tree

and the following code to draw these:

    tree = random_tree(32)
    tree.plot(vertex_size = 1, vertex_labels = False, graph_border = True, iterations = 4096)

  [1]: http://www.freeimagehosting.net/ta8ni
  [2]: http://www.freeimagehosting.net/3yl9z
  [3]: http://www.freeimagehosting.net/qqcm4