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happy-sanhappy-san
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Add illustrations for cases of deletion from left subtree.
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‎lib/trees/avl_tree.dart‎

Lines changed: 128 additions & 76 deletions
Original file line numberDiff line numberDiff line change
@@ -306,60 +306,112 @@ class AvlTree<V extends Comparable> extends BinaryTreeADT<AvlNode<V>, V> {
306306
// Right subtree of [node]
307307
var rChild = node.right;
308308

309-
// [rChild] was balanced.
310-
// N
311-
// \
312-
// R
313-
// / \
314-
// ** **
315-
if (rChild!.balanceFactor == 0) {
316-
// Single left rotation about [node] is performed to balance it.
317-
node = _rotateLeft(node);
309+
switch (rChild!.balanceFactor) {
318310

319-
node.balanceFactor = -1;
320-
rChild.balanceFactor = 1;
321-
_isShorter = false;
322-
}
311+
// [rChild] was balanced, single left rotation about [node] is performed
312+
// to balance the [node].
313+
case 0:
314+
//
315+
// N N R
316+
// / \ /↶\ / \
317+
// ...█...R.... ⟶ h ...█...R.... ⟶ ....N...█...
318+
// h+1 ...▒../.\... ....../.\... .../.\..█.. h+1
319+
// .....█...█.. .....█...█.. h ..█...█.....
320+
// h+1 ..█...█.. h+1 h+1 ..█...█.. h+1 ......█.. h+1
321+
//
322+
node = _rotateLeft(node);
323323

324-
// [rChild] was right heavy.
325-
// N
326-
// \
327-
// R
328-
// / \
329-
// * **
330-
else if (rChild.balanceFactor == -1) {
331-
node.balanceFactor = rChild.balanceFactor = 0;
332-
// Single left rotation about [node] is performed to balance it.
333-
node = _rotateLeft(node);
334-
}
324+
node.balanceFactor = -1;
325+
rChild.balanceFactor = 1;
335326

336-
// [rChild] was left heavy.
337-
else {
338-
// Left subtree of [rChild].
339-
var lGrandChild = rChild.left;
340-
switch (lGrandChild!.balanceFactor) {
327+
// Height of the subtree rooted at N does not change, hence [_isShorter]
328+
// is made `false`.
329+
_isShorter = false;
330+
break;
341331

342-
// [lGrandChild] was balanced.
343-
case 0:
344-
node.balanceFactor = rChild.balanceFactor = 0;
345-
break;
332+
// [rChild] was right heavy, single left rotation about [node] is
333+
// performed to balance the [node].
334+
case -1:
335+
//
336+
// N N R
337+
// / \ /↶\ / \
338+
// ...█...R.... ⟶ h ...█...R.... ⟶ ....N...█...
339+
// h+1 ...▒../.\... ....../.\... .../.\..█.. h+1
340+
// .....█...█.. .. h █...█.. h ..█...█.....
341+
// ... h ...█.. h+1 .........█.. h+1 ...... h ...
342+
//
343+
node = _rotateLeft(node);
346344

347-
// [lGrandChild] was left heavy.
348-
case 1:
349-
node.balanceFactor = 0;
350-
rChild.balanceFactor = -1;
351-
break;
345+
node.balanceFactor = rChild.balanceFactor = 0;
352346

353-
// [lGrandChild] was right heavy.
354-
case -1:
355-
node.balanceFactor = 1;
356-
rChild.balanceFactor = 0;
357-
break;
358-
}
359-
lGrandChild.balanceFactor = 0;
360-
// Right Left rotation is perfomed to balance [node].
361-
node.right = _rotateRight(rChild);
362-
node = _rotateLeft(node);
347+
// Height of the subtree rooted at N has changed, hence [_isShorter]
348+
// remains `true`.
349+
break;
350+
351+
// [rChild] was left heavy. In this case, a single rotation will not
352+
// suffice so double rotation is performed. L is the root
353+
// node of [rChild]'s left subtree.
354+
//
355+
// N N
356+
// / \ / \
357+
// ...█...R.... ...█...R....
358+
// h+1 ...▒../.\... h+1 ...▒. /.\...
359+
// .....█...█.. h .... L ..█.. h
360+
// .h+1 █...... / \
361+
//
362+
case 1:
363+
364+
// Left subtree of [rChild].
365+
var lGrandChild = rChild.left;
366+
367+
switch (lGrandChild!.balanceFactor) {
368+
369+
// [lGrandChild] was balanced.
370+
case 0:
371+
// N N N
372+
// / \ / \ /↶\ L
373+
// ...█...R.... h ...█...R.... h ...█...L...... / \
374+
// h+1 ...▒. /.\... ⟶ ..... /↷\... ⟶ ....../.\..... ⟶ N R
375+
// .... L ..█.. h .... L ..█.. h h..█...R.... / \ / \
376+
// / \ / \ ...../.\... h..█...█.█...█..h
377+
// h ..█...█.. h h ..█...█.. h h..█...█.. h
378+
//
379+
node.balanceFactor = rChild.balanceFactor = 0;
380+
break;
381+
382+
// [lGrandChild] was left heavy.
383+
case 1:
384+
// N N N
385+
// / \ / \ /↶\ L
386+
// ...█...R.... h ...█...R.... h ...█...L...... / \
387+
// h+1 ...▒. /.\... ⟶ ..... /↷\... ⟶ ....../.\..... ⟶ N R
388+
// .... L ..█.. h .... L ..█.. h h..█...R.... / \ / \
389+
// / \ / \ ...../.\... h..█...█ ▀ █..h
390+
// h ..█...▀.. h-1 h ..█...▀.. h-1 h-1..▀...█.. h │
391+
// h-1 ┘
392+
node.balanceFactor = 0;
393+
rChild.balanceFactor = -1;
394+
break;
395+
396+
// [lGrandChild] was right heavy.
397+
case -1:
398+
// N N N
399+
// / \ / \ /↶\ L
400+
// ...█...R.... h ...█...R.... h ...█...L...... / \
401+
// h+1 ...▒. /.\... ⟶ ..... /↷\... ⟶ ....../.\..... ⟶ N R
402+
// .... L ..█.. h .... L ..█.. h h-1..▀...R.... / \ / \
403+
// / \ / \ ...../.\... h..█ ▀ █...█..h
404+
// h-1 ..▀...█.. h h-1 ..▀...█.. h h..█...█.. h │
405+
// h-1 ┘
406+
node.balanceFactor = 1;
407+
rChild.balanceFactor = 0;
408+
break;
409+
}
410+
lGrandChild.balanceFactor = 0;
411+
// Right Left rotation is perfomed to balance [node].
412+
node.right = _rotateRight(rChild);
413+
node = _rotateLeft(node);
414+
break;
363415
}
364416
return node;
365417
}
@@ -451,47 +503,47 @@ class AvlTree<V extends Comparable> extends BinaryTreeADT<AvlNode<V>, V> {
451503
}
452504

453505
/// Updates [balanceFactor] of the [node] when deletion is done from it's left
454-
/// subtree.\
506+
/// subtree.
455507
AvlNode<V> _dUpdateLeftBalanceFactor(AvlNode<V> node) {
456508
switch (node.balanceFactor) {
457509

458-
// [node] was balanced before deletion. (Both left and right subtree had
459-
// the same height.)
460-
// N
461-
// / \
462-
// ** **
510+
// [node] was balanced before deletion, is right heavy now.
463511
case 0:
464-
// [node] is right heavy now.
465-
// N
466-
// / \
467-
// * **
512+
// N N
513+
// / \ ⟶ / \
514+
// ...█...█... h ...█...█...
515+
// h+1 ...▒...█... h+1 .......█... h+1
516+
//
468517
node.balanceFactor = -1;
518+
519+
// Balance factors of the ancestors of this [node] will remain
520+
// unchanged (since height of the subtree rooted at N didn't change).
521+
// Therefore [_isShorter] is set to `false`.
469522
_isShorter = false;
470523
break;
471524

472-
// [node] was left heavy before deletion. (Left subtree's height was one
473-
// more than that of right subtree.)
474-
// N
475-
// / \
476-
// ** *
525+
// [node] was left heavy before deletion, is balanced now.
477526
case 1:
478-
// [node] is balanced now.
479-
// N
480-
// / \
481-
// * *
527+
// N N
528+
// / \ ⟶ / \
529+
// ...█...█... h h ...█...█... h
530+
// h+1 ...▒....... ...........
531+
//
482532
node.balanceFactor = 0;
533+
534+
// Height of the subtree rooted at N has decreased, [_isShorter] remains
535+
// `true`.
483536
break;
484537

485-
// [node] was right heavy before deletion. (Right subtree's height was one
486-
// more than that of left subtree.)
487-
// N
488-
// / \
489-
// * **
538+
// [node] was right heavy before deletion, is imbalanced now, has to be
539+
// balanced.
490540
case -1:
491-
// [node] is imbalanced now, has to be balanced.
492-
// N
493-
// \
494-
// **
541+
// N N
542+
// / \ / \
543+
// ...█...█... ⟶ h-1 ...█...█...
544+
// h ...▒...█... .......█...
545+
// .......█... h+1 .......█... h+1
546+
//
495547
node = _dBalanceRightHeavy(node);
496548
}
497549
return node;

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