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

Over $32$ nodes: The random tree over 32 nodes. http://www.freeimagehosting.net/newuploads/ta8ni.png

Over $96$ nodes: The random tree over 96 nodes. http://www.freeimagehosting.net/newuploads/3yl9z.png

Over $512$ nodes: The random tree over 512 nodes. http://www.freeimagehosting.net/newuploads/qqcm4.png

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 = 24, vertex_labels = False, graph_border = True, iterations = 4096)