The AVL Tree Rotations Tutorial

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1 The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk of writing n AVL tree lss in Jv, I ws left souring the we for useful informtion on how this ll works. There ws lot of useful informtion on the wikipedi pges for AVL tree nd Tree rottion. You n find links to these pges in setion 4. The tree rottion pge on wikipedi is lking, I feel. The AVL tree pge needs work s well, ut this pge is hurting dly, nd t some point in the future, I will likely integrte most of this doument into tht pge. This doument overs oth types of rottions, nd ll 4 pplitions of them. There is lso smll setion on deiding whih rottions to use in different situtions. 1. Rottions: How they work A tree rottion n e n imtimidting onept t first. You end up in sitution where you're juggling nodes, nd these nodes hve trees tthed to them, nd it n ll eome onfusing very fst. I find it helps to lok out wht's going on with ny of the sutrees whih re tthed to the nodes you're fumling with, ut tht n e hrd. Left Rottion (LL) Imgine we hve this sitution: Figure 1-1 To fix this, we must perform left rottion, rooted t A. This is done in the following steps: eomes the new root. tkes ownership of 's left hild s its right hild, or in this se, null. tkes ownership of s its left hild. The tree now looks like this: Figure 1-2 Right Rottion (RR) A right rottion is mirror of the left rottion opertion desried ove. Imgine we hve this sitution: Figure 1-3

2 To fix this, we will perform single right rottion, rooted t C. This is done in the following steps: eomes the new root. tkes ownership of 's right hild, s its left hild. In this se, tht vlue is null. tkes ownership of, s it's right hild. The resulting tree: Figure 1-4 Left-Right Rottion (LR) or "Doule left" Sometimes single left rottion is not suffiient to lne n unlned tree. Tke this sitution: Figure 1-5 Perfet. It's lned. Let's insert ''. Figure 1-6 Our initil retion here is to do single left rottion. Let's try tht. Figure 1-7 Our left rottion hs ompleted, nd we're stuk in the sme sitution. If we were to do single right rottion in this sitution, we would e right k where we strted. Wht's using this? The nswer is tht this is result of the right sutree hving negtive lne. In other words, euse the right sutree ws left hevy, our rottion ws not suffiient. Wht n we do? The nswer is to perform right rottion on the right sutree. Red tht gin. We will perform right rottion on the right sutree. We re not rotting on our urrent root. We re rotting on our right hild. Think of our right sutree, isolted from our min tree, nd perform right rottion on it: Before:

3 Figure 1-8 After: Figure 1-9 After performing rottion on our right sutree, we hve prepred our root to e rotted left. Here is our tree now: Figure 1-10 Looks like we're redy for left rottion. Let's do tht: Figure 1-11 Voil. Prolem solved. Right-Left Rotition (RL) or "Doule right" A doule right rottion, or right-left rottion, or simply RL, is rottion tht must e performed when ttempting to lne tree whih hs left sutree, tht is right hevy. This is mirror opertion of wht ws illustrted in the setion on Left-Right Rottions, or doule left rottions. Let's look t n exmple of sitution where we need to perform Right-Left rottion. Figure 1-12 In this sitution, we hve tree tht is unlned. The left sutree hs height of 2, nd the right sutree hs height of 0. This mkes the lne ftor of our root node,, equl to -2. Wht do we do? Some kind of right rottion is lerly neessry, ut single right rottion will not solve our prolem. Let's try it: Figure 1-13

4 Looks like tht didn't work. Now we hve tree tht hs lne of 2. It would pper tht we did not omplish muh. Tht is true. Wht do we do? Well, let's go k to the originl tree, efore we did our pointless right rottion: Figure 1-14 The reson our right rottion did not work, is euse the left sutree, or '', hs positive lne ftor, nd is thus right hevy. Performing right rottion on tree tht hs left sutree tht is right hevy will result in the prolem we just witnessed. Wht do we do? The nswer is to mke our left sutree left-hevy. We do this y performing left rottion our left sutree. Doing so leves us with this sitution: Figure 1-15 This is tree whih n now e lned using single right rottion. We n now perform our right rottion rooted t C. The result: Figure 1-16 Blne t lst. 2. Rottions, When to Use Them nd Why How to deide when you need tree rottion is usully esy, ut determining whih type of rottion you need requires little thought. A tree rottion is neessry when you hve inserted or deleted node whih leves the tree in n unlned stte. An unlned stte is defined s stte in whih ny sutree hs lne ftor of greter thn 1, or less thn -1. Tht is, ny tree with differene etween the heights of its two sutrees greter thn 1, is onsidered unlned. This is lned tree: Figure 2-1 1

5 2 3 This is n unlned tree: Figure This tree is onsidered unlned euse the root node hs lne ftor of 2. Tht is, the right sutree of 1 hs height of 2, nd the height of 1's left sutree is 0. Rememer tht lne ftor of tree with left sutree A nd right sutree B is B - A Simple. In figure 2-2, we see tht the tree hs lne of 2. This mens tht the tree is onsidered "right hevy". We n orret this y performing wht is lled "left rottion". How we determine whih rottion to use follows few si rules. See psuedo ode: IF tree is right hevy IF tree's right sutree is left hevy Perform Doule Left rottion ELSE Perform Single Left rottion ELSE IF tree is left hevy IF tree's left sutree is right hevy Perform Doule Right rottion ELSE Perform Single Right rottion As you n see, there is sitution where we need to perform "doule rottion". A single rottion in the situtions desried in the pseudo ode leve the tree in n unlned stte. Follow these rules, nd you should e le to lne n AVL tree following n insert or delete every time.

6 3. Summry It s importnt to understnd tht the exmples ove were on very smll trees to keep the onepts ler. In theory, however, if you develop n pplition whih uses AVL trees, progrmming for the situtions shown ove while using the rules provided should sle just fine. If you hve omments, questions or ritiisms, feel free to e-mil me t 4. Further Reding - Tree rottion pge on Wikipedi, - AVL tree pge on Wikipedi, - Animted AVL Tree Jv pplet, - AVL Trees: Tutoril nd C++ Implementtion,

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