English: This is a graph, generated in bottom-up fashion, of the orbits of all numbers under the Collatz map with an orbit length of 20 or less.

Created with Graphviz, with the help of this Python program:

 # This python script generates a graph that shows 20 levels of the Collatz Conjecture.
 
 level = [1] # The root of the tree contains only "1"
 results = {} # This will hold the relationships for the graph
 
 # Loop 20 times
 for c in range(20):
     newlevel = set() # This will be the set of numbers in the level above us
     for x in level: # We now loop through each number in our current level
 
         # There will only sometimes be an odd multiple of 3 (plus one) above us
         # When (x-1) mod 3 is 0, x is an even number half the time
         # So we check against (x-4) mod 6 because we don't want to find evens
         odd = (x-1)/3 if not (x-4)%6 else 0
         if odd > 1: # We don't want 1 (the root) or 0 (no odd found)
             newlevel.add(odd) # Put this into the next level
             results[odd] = x # Add a relationship to the graph
 
         newlevel.add(x*2); # There will always be an even number above us
         results[x*2] = x # Add a relationship to the graph
 
     # Finally, advance to the next level
     level = newlevel
 
 # Technically, 1 is odd so 1*3+1 = 4. If included though, this tends to make 2 and 1 overlap on the graph, which is ugly.
 # Uncomment to include this relationship:
 # results[1] = 4;
 
 # Now we write the graph out to a file in dot format so that graphviz can read it
 f = open('collatz.dot', 'w')
 # For aesthetic reasons we set 8 as the centre node instead of 1
 f.write('digraph G {\n    size="16,16";\n    root=8;\n    splines=true;\n')
 for x in results:
     f.write("    %d -> %d;\n" % (x, results[x]))
 f.write('}\n')
 f.close()
 print "Generated %d nodes" % len(results)