add comment for btree

Author: xtaci

include/btree.h

@@ -231,32 +231,52 @@ namespace alg {
 i = i - 1;
 }
 
-if (x->key[i] == k) {// key in this node.
-if (x->flag & LEAF) {// if it's a leaf node. case 1.
+if (x->key[i] == k) {// key exists in this node.
+// case 1.
+// If the key k is in node x and x is a leaf, delete the key k from x.
+if (x->flag & LEAF) {
 int j;
-for (j = i;j<x->n-1;j++) {// shift copy
+for (j = i;j<x->n-1;j++) {// shifting the keys.
 x->key[j] = x->key[j+1];
 }
 WRITE(x);
 return;
-} else { // in non-leaf node
+} else {
+// case 2a:
+// If the child y that precedes k in node x has at least t 
+// keys, then find the predecessor k0 of k in the subtree 
+// rooted at y. Recursively delete k0, and replace k by k0 in x. 
+// (We can find k0 and delete it in a single downward pass.)
 std::auto_ptr<node_t> y(READ(x, i));
-if (y->n >= T) {// case 2a.
-x->key[i] = y->key[y->n-1];
+if (y->n >= T) {
+int32_t k0 = y->key[y->n-1];
+x->key[i] = k0;
 WRITE(x);
-delete_op(y.get(), x->key[i]);
+delete_op(y.get(), k0);
 return;
 }
-
+
+// case 2b.
+// If y has fewer than t keys, then, symmetrically, examine 
+// the child z that follows k in node x. If z has at least t keys,
+// then find the successor k0 of k in the subtree rooted at z. 
+// Recursively delete k0, and replace k by k0 in x. (We can find k0 
+// and delete it in a single downward pass.)
 std::auto_ptr<node_t> z(READ(x, i+1));
-if (z->n >= T) {// case 2b.
-x->key[i] = z->key[0];
+if (z->n >= T) {
+int32_t k0 = z->key[0];
+x->key[i] = k0;
 WRITE(x);
-delete_op(z.get(), x->key[i]);
+delete_op(z.get(), k0);
 return;
 }
 
-if (y->n == T-1 && z->n == T-1) { // case 2c
+// case 2c:
+// Otherwise, if both y and ´ have only t 2 1 keys, 
+// merge k and all of ´ into y, so that x loses both k and the 
+// pointer to ´, and y now contains 2t c 1 keys. 
+// Then free ´ and recursively delete k from y.
+if (y->n == T-1 && z->n == T-1) {
 // merge k & z into y
 y->key[y->n] = k;