问题
I am new to recursion and trying to understand this code snippet. I'm studying for an exam, and this is a "reviewer" I found from Standford' CIS Education Library (From Binary Trees by Nick Parlante).
I understand the concept, but when we're recursing INSIDE THE LOOP, it all blows! Please help me. Thank you.
countTrees() Solution (C/C++)
/*
For the key values 1...numKeys, how many structurally unique
binary search trees are possible that store those keys.
Strategy: consider that each value could be the root.
Recursively find the size of the left and right subtrees.
*/
int countTrees(int numKeys) {
if (numKeys <=1) {
return(1);
}
// there will be one value at the root, with whatever remains
// on the left and right each forming their own subtrees.
// Iterate through all the values that could be the root...
int sum = 0;
int left, right, root;
for (root=1; root<=numKeys; root++) {
left = countTrees(root - 1);
right = countTrees(numKeys - root);
// number of possible trees with this root == left*right
sum += left*right;
}
return(sum);
}
回答1:
Imagine the loop being put "on pause" while you go in to the function call.
Just because the function happens to be a recursive call, it works the same as any function you call within a loop.
The new recursive call starts its for loop and again, pauses while calling the functions again, and so on.
回答2:
- For recursion, it's helpful to picture the call stack structure in your mind.
- If a recursion sits inside a loop, the structure resembles (almost) a N-ary tree.
- The loop controls horizontally how many branches at generated while the recursion decides the height of the tree.
- The tree is generated along one specific branch until it reaches the leaf (base condition) then expand horizontally to obtain other leaves and return the previous height and repeat.
I find this perspective generally a good way of thinking.
回答3:
Look at it this way: There's 3 possible cases for the initial call:
numKeys = 0
numKeys = 1
numKeys > 1
The 0 and 1 cases are simple - the function simply returns 1 and you're done. For numkeys 2, you end up with:
sum = 0
loop(root = 1 -> 2)
root = 1:
left = countTrees(1 - 1) -> countTrees(0) -> 1
right = countTrees(2 - 1) -> countTrees(1) -> 1
sum = sum + 1*1 = 0 + 1 = 1
root = 2:
left = countTrees(2 - 1) -> countTrees(1) -> 1
right = countTrees(2 - 2) -> countTrees(0) -> 1
sum = sum + 1*1 = 1 + 1 = 2
output: 2
for numKeys = 3:
sum = 0
loop(root = 1 -> 3):
root = 1:
left = countTrees(1 - 1) -> countTrees(0) -> 1
right = countTrees(3 - 1) -> countTrees(2) -> 2
sum = sum + 1*2 = 0 + 2 = 2
root = 2:
left = countTrees(2 - 1) -> countTrees(1) -> 1
right = countTrees(3 - 2) -> countTrees(1) -> 1
sum = sum + 1*1 = 2 + 1 = 3
root = 3:
left = countTrees(3 - 1) -> countTrees(2) -> 2
right = countTrees(3 - 3) -> countTrees(0) -> 1
sum = sum + 2*1 = 3 + 2 = 5
output 5
and so on. This function is most likely O(n^2), since for every n keys, you're running 2*n-1 recursive calls, meaning its runtime will grow very quickly.
回答4:
Just to remember that all the local variables, such as numKeys, sum, left, right, root are in the stack memory. When you go to the n-th depth of the recursive function , there will be n copies of these local variables. When it finishes executing one depth, one copy of these variable will be popped up from the stack.
In this way, you will understand that, the next-level depth will NOT affect the current-level depth local variables (UNLESS you are using references, but we are NOT in this particular problem).
For this particular problem, time-complexity should be carefully paid attention to. Here are my solutions:
/* Q: For the key values 1...n, how many structurally unique binary search
trees (BST) are possible that store those keys.
Strategy: consider that each value could be the root. Recursively
find the size of the left and right subtrees.
http://stackoverflow.com/questions/4795527/
how-recursion-works-inside-a-for-loop */
/* A: It seems that it's the Catalan numbers:
http://en.wikipedia.org/wiki/Catalan_number */
#include <iostream>
#include <vector>
using namespace std;
// Time Complexity: ~O(2^n)
int CountBST(int n)
{
if (n <= 1)
return 1;
int c = 0;
for (int i = 0; i < n; ++i)
{
int lc = CountBST(i);
int rc = CountBST(n-1-i);
c += lc*rc;
}
return c;
}
// Time Complexity: O(n^2)
int CountBST_DP(int n)
{
vector<int> v(n+1, 0);
v[0] = 1;
for (int k = 1; k <= n; ++k)
{
for (int i = 0; i < k; ++i)
v[k] += v[i]*v[k-1-i];
}
return v[n];
}
/* Catalan numbers:
C(n, 2n)
f(n) = --------
(n+1)
2*(2n+1)
f(n+1) = -------- * f(n)
(n+2)
Time Complexity: O(n)
Space Complexity: O(n) - but can be easily reduced to O(1). */
int CountBST_Math(int n)
{
vector<int> v(n+1, 0);
v[0] = 1;
for (int k = 0; k < n; ++k)
v[k+1] = v[k]*2*(2*k+1)/(k+2);
return v[n];
}
int main()
{
for (int n = 1; n <= 10; ++n)
cout << CountBST(n) << '\t' << CountBST_DP(n) <<
'\t' << CountBST_Math(n) << endl;
return 0;
}
/* Output:
1 1 1
2 2 2
5 5 5
14 14 14
42 42 42
132 132 132
429 429 429
1430 1430 1430
4862 4862 4862
16796 16796 16796
*/
回答5:
You can think of it from the base case, working upward.
So, for base case you have 1 (or less) nodes. There is only 1 structurally unique tree that is possible with 1 node -- that is the node itself. So, if numKeys is less than or equals to 1, just return 1.
Now suppose you have more than 1 key. Well, then one of those keys is the root, some items are in the left branch and some items are in the right branch.
How big are those left and right branches? Well it depends on what is the root element. Since you need to consider the total amount of possible trees, we have to consider all configurations (all possible root values) -- so we iterate over all possible values.
For each iteration i, we know that i is at the root, i - 1 nodes are on the left branch and numKeys - i nodes are on the right branch. But, of course, we already have a function that counts the total number of tree configurations given the number of nodes! It's the function we're writing. So, recursive call the function to get the number of possible tree configurations of the left and right subtrees. The total number of trees possible with i at the root is then the product of those two numbers (for each configuration of the left subtree, all possible right subtrees can happen).
After you sum it all up, you're done.
So, if you kind of lay it out there's nothing special with calling the function recursively from within a loop -- it's just a tool that we need for our algorithm. I would also recommend (as Grammin did) to run this through a debugger and see what is going on at each step.
回答6:
Each call has its own variable space, as one would expect. The complexity comes from the fact that the execution of the function is "interrupted" in order to execute -again- the same function. This code:
for (root=1; root<=numKeys; root++) {
left = countTrees(root - 1);
right = countTrees(numKeys - root);
// number of possible trees with this root == left*right
sum += left*right;
}
Could be rewritten this way in Plain C:
root = 1;
Loop:
if ( !( root <= numkeys ) ) {
goto EndLoop;
}
left = countTrees( root -1 );
right = countTrees ( numkeys - root );
sum += left * right
++root;
goto Loop;
EndLoop:
// more things...
It is actually translated by the compiler to something like that, but in assembler. As you can see the loop is controled by a pair of variables, numkeys and root, and their values are not modified because of the execution of another instance of the same procedure. When the callee returns, the caller resumes the execution, with the same values for all values it had before the recursive call.
回答7:
IMO, key element here is to understand function call frames, call stack, and how they work together.
In your example, you have bunch of local variables which are initialised but not finalised in the first call. It's important to observe those local variables to understand the whole idea. At each call, the local variables are updated and finally returned in a backwards manner (most likely it's stored in a register before each function call frame is popped off from the stack) up until it's added to the initial function call's sum variable.
The important distinction here is - where to return. If you need accumulated sum value like in your example, you cannot return inside the function which would cause to early-return/exit. However, if you depend on a value to be in a certain state, then you can check if this state is hit inside the for loop and return immediately without going all the way up.
来源：https://stackoverflow.com/questions/4795527/how-recursion-works-inside-a-for-loop