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Here (click the image for a higherres version) is a picture of a branching Brownian motion in one dimension (the xaxis represents time, and each particle is a different colour): Drawing discrete trees in a useful way is tricky, as the number of possible leaves in each generation increases exponentially. Here are a couple of attempts at drawing a random binary search tree (random quicksort) with 20,000 insertions. Colour represents age (red means the node was added early on, blue late). Click for higherres.
And for even higher res, click below, but beware, it's 2.5mb.
One interesting quantity in the random binary search tree is the number of leaves at the topmost level of the tree (called the "fringe" in the article mentioned above). The picture below shows this quantity (thick blue line) together with the number at the next level down (thin red line) and the level below that (dashed green line). It also features Rolf Harris, but you needn't worry about him. You can see the picture in higher resulotion by clicking on it. The horizontal axis represents time, and the vertical axis is the number of particles present.
I also drew some prettier, but less useful, pictures of the BST. Below, the top level (the fringe) is orange/red, the level below that is blue/purple, and the third level is green  and it was run over a much shorter timescale. Again, click the image for the fullsized version.
The following picture is from a simulation of a model that I'm working on with Marcel Ortgiese. More information will follow when we have something concrete to share!
I can't take any credit for the next model  see instead Jeff Steif's survey and references within  but I was experimenting with a programming language called Processing one day and drew some pictures of dynamical percolation.
Click here to see the program in action (and get the source code if you want it)  it requires Java. Also click here for a smaller, fastermoving version. For those who know what the following means, it's just critical bond percolation on Z^2 where each bond randomly updates its status at independent exponential times. The clusters belonging to the origin and four other evenlyspaced points (one in each corner) are shown in different colours. The important thing to note from a mathematical point of view is that "large" clusters (of each colour) appear and disappear very suddenly. The following is a twodimensional version of something that I have starting thinking about in one dimension (on the circle). Onedimensional pictures tend to be less interesting so I drew this instead. (In fact it's not even quite the 2d equivalent of the 1d model I have been looking at, but it's similar.) In each timestep each square has a chance to fragment into four smaller squares, or  if there are three other squares of the same size available  to combine with three others to make a bigger square. Every time a fragmentation or coagulation occurs the centres of the newlycreated squares are distributed uniformly over the window. Where I say "has a chance", you might like to ask yourself what probabilities one should choose to get interesting questions. Click the image to see the program in action  note that it doesn't start in equilibrium so you have to wait a while for it to get going.
Here is a picture of a branching Brownian motion on a circle (distance from the origin represents time). It's hard to see what's going on  but it's kind of pretty I think.
And another picture of BBM on a circle: this time it's in 3D, with the Zaxis representing time. Slightly easier to see what's actually happening, at least to start off with.
