Understanding and visualizing recursion

Question

I referred to several questions here about recursion but I am not able to understand how recursion works for this particular problem: Recursive program to get all combination of characters in a string in Python:

st= []
def combi(prefix, s):
    if len(s)==0: return 
    else:
        st.append(prefix+s[0])        

        ''' printing values so that I can see what happens at each stage '''
        print "s[0]=",s[0]
        print "s[1:]=",s[1:]
        print "prefix=",prefix
        print "prefix+s[0]=",prefix+s[0]
        print "st=",st

        combi(prefix+s[0],s[1:])
        combi(prefix,s[1:])
        return st

print combi("",'abc')

I've made it print the values so that I can see what's happening. This is the output:

s[0]= a
s[1:]= bc
prefix= 
prefix+s[0]= a
st= ['a']
s[0]= b
s[1:]= c
prefix= a
prefix+s[0]= ab
st= ['a', 'ab']
s[0]= c
s[1:]= 
prefix= ab
prefix+s[0]= abc
st= ['a', 'ab', 'abc']
s[0]= c
s[1:]= 
prefix= a  ----> How did prefix become 'a' here. Shouldn't it be 'abc' ? 
prefix+s[0]= ac
st= ['a', 'ab', 'abc', 'ac']
.........
.........
['a', 'ab', 'abc', 'ac', 'b', 'bc', 'c'] # final output

Full output: http://pastebin.com/Lg3pLGtP

As I've shown in the output, how did prefix become 'ab'?

I tried to visualize the recursive calls for the combi(prefix+s[0],s[1:]). Am I understanding it right?

Answer 1

Theres a python module for that

Generated with:

from rcviz import callgraph, viz
st= []
@viz
def combi(prefix, s):
    if len(s)==0: 
        return 
    else:
        st.append(prefix+s[0])     
        combi.track(st = st) #track st in rcviz 
        combi(prefix+s[0],s[1:])
        combi(prefix,s[1:])
        return st

print combi("",'abc')
callgraph.render("combi.png")

Answer 2

There are two recursive calls to combi() in the function. Thus the path of calls is not a single line, but rather a binary tree that forks. What you are seeing is the second half of the tree.

Answer 3

I drew the recursion tree. By Depth First Traversal, the final output is got at the last node. This visualization helps understand what's happening.

Answer 4

I've written a python package called recursion-visualiser which helps to draw a recursion tree for any arbitary recursive function. You have to simply add a decorator and boom you have nice animation and recursion tree saved as gif and png.

from visualiser.visualiser import Visualiser as vs

st= []
@vs(show_argument_name=False, node_properties_kwargs={"shape":"record", 
"color":"#f57542", "style":"filled", "fillcolor":"grey"})
def combi(prefix, s):
    if len(s) == 0:
        return " "
    else:
        st.append(prefix + s[0])
        combi(prefix=prefix + s[0], s=s[1:])
        combi(prefix=prefix, s=s[1:])
        return st

print(combi(prefix="",s='abc'))
vs.make_animation("combinations.gif", delay=3)

Here is the output gif:
Also, a recursion tree saved as png with return value:

Check out more examples here.