In this post I will try to review some of the main versions of self balancing binary trees, provide Java implementations and micro benchmark their performance under various conditions.

Why do we need binary search trees?

Lets say we have a list as an input and we need to find some element in it. What is the best way to do it? Naive implementation could be simply iterating through all the elements and check each one. That maybe be fine for small lists and simple use cases. But can we do better? Probably not with a simple random array. However if we sort it beforehand it is possible to “divide and conqueror”. That is we can check element in the middle and if the one we are looking for is larger we do the same with sub list n/2 to n (where n is number of elements in the list). So we cut through search space considerably (O(log n) complexity instead of O(n)). That is fine but we need to sort the input before with some sorting algorithm. And if we are constantly modifying (inserting/deleting) the input and searching through it, it’s not practical to sort it each time before search. Here is where binary search trees steps in. We modify the way data is kept. Not in a simple list/array but in a tree like data structure. Looking at the picture you probably already see why it is easier to search in such a structure.

However binary search tree can get unbalanced and then loose its efficiency. To solve that problem self balancing binary search trees were invented. I am not going to explain how they work in detail here just provide general information and some ideas how they can be used. Also I will provide my implementations in java and some benchmarks so you can start playing with it if you want.

AVL tree

One of the oldest, most well known and most popular tree data structure. It saves additional height information in each node and re balances tree if height of one node is higher than its sibling by 2. That keeps tree rigidly balanced so search is extremely fast on AVL tree. However that comes with a cost of rebalancing and that can make insert/delete operations slightly more expensive than other trees.

Quick link to my implementation on GitHub

Red-Black tree

Most widely used self balancing binary search tree. It also saves additional information on nodes (black node or red node) and preserves tree balance based on certain rules about how nodes are colored. It can be less balanced than AVL tree (which means slower search) but with more efficient inserts and deletes. Also because only two colors required, Red-Black tree needs only one bit of additional information per node. All of this makes it most popular choice for general use cases. Java TreeSet is implemented as Red-Black tree.

Quick link to my implementation on GitHub

Scapegoat tree

This tree has advantage that it doesn’t need any additional memory per node. Once it finds out that the tree has to be re balanced it finds a scapegoat node and performs re balance operation on it. So it might be a good choice when memory is an issue. To find out if the tree is out of balance it uses parameter alpha. This allows to parametrise a scapegoat tree with different values of alpha which means different degrees of balance which means a choice between faster search or faster inserts. Imagine doing that dynamically – when there is heavy load on searches and not too much on inserts/deletes, you can lower alpha and increase it when amount of inserts/deletes increases.

Quick link to my implementation on GitHub

Treap

Basically randomized binary search tree and heap hybrid. It randomly generates and saves additional node parameter priority. Tree is then balanced such as priority values satisfies heap property. It is easy to implement and I think has a possibility of fast set operations (I did not implement it in my code). Other than that I don’t know if it has any advantage over other trees. Maybe instead of random priority flag we can use a non random (somehow calculated) importance flag which would keep more important nodes at the top?

Quick link to my implementation on GitHub

Splay tree

Splay tree is pretty interesting. It performs operation called splaying not only on inserts and deletes, but also when searching. This operation performs tree rotations and brings element up to the top. It means that recently accessed elements are always near the top. You can see how it can be very useful in certain use cases like some cache implementations. However additional rotations on search adds additional overhead and if each element is as likely to be searched then splaying doesn’t do that much good.

Quick link to my implementation on GitHub

Short summary

First choice for BST is a Red-Black tree as it is most universal data structure. However if your tree will be used mostly for search then take a look at AVL tree. If mostly the same elements are retrieved again and again then Splay tree can be even better. If sometimes there is a heavy read and sometimes a heavy write load then Scapegoat tree with different parameters alpha might be helpful. If for some reason you need to implement tree yourself and don’t yet feel very confident about your coding skills – Treap might be you choice.

Benchmarks

Here I have prepared some simple benchmarks for my own implementations of each tree. Please keep in mind that I might have made some mistakes or my implementation can be inefficient so let me know if you find that to be the case. Keep in mind that ms/op in units column means whole 100k elements test in ms. I’ve used jmh framework for benchmarking.

This first benchmark inserts elements from 1 to 100K. We see that all scapegoat trees performed very poorly. Not sure why. Splay tree performed phenomenally, however it is not that good once you realize that tree is not balanced at all and in fact is basically a linked list. Interesting that AVL tree was much better than Red-Black tree. Treap was quite good too (maybe even the best choice as it is much simpler than only slightly faster AVL tree). Another interesting observation is that less balanced Scapegoat tree (0.9 alpha) had slower inserts than 0.75 alpha balanced Scapegoat tree. In this case it is no surprise because we insert sorted elements. Having tree re balance too rarely in this situation increases costs for finding next element’s place once tree grows larger.

Insert sorted elements benchmark Score Score error Units AVL tree 9.420 0.270 ms/op JDK TreeSet 13.981 0.479 ms/op Red-Black tree 17.430 0.917 ms/op Scapegoat tree (alpha 0.6) 131.432 4.159 ms/op Scapegoat tree (alpha 0.75) 76.769 2.538 ms/op Scapegoat tree (alpha 0.9) 77.999 1.775 ms/op Splay tree 1.229 0.039 ms/op Treap 9.576 1.102 ms/op

Once again lets insert 100k elements but this time they are random and not sorted. My AVL tree implementation did quite well and Red-Black tree implementation was much slower than we could expect (remember, it suppose to be faster on inserts). Maybe my implementation is not very efficient or I made a mistake somewhere. Red-Black tree implementation from JDK was fastest here (but only barely faster than my non optimized AVL tree) and Splay tree was slowest so no more surprises.

Insert random elements benchmark Score Score error Units AVL tree 39.051 1.998 ms/op JDK TreeSet 38.811 4.356 ms/op Red-Black tree 59.052 7.871 ms/op Scapegoat tree (alpha 0.6) 50.738 4.400 ms/op Scapegoat tree (alpha 0.75) 44.077 1.192 ms/op Scapegoat tree (alpha 0.9) 44.862 2.168 ms/op Splay tree 76.457 4.114 ms/op Treap 47.978 1.656 ms/op

This next benchmark inserts 100K elements and then deletes them one by one. One more time my implementation of Red-Black tree under performs (any help why is that would be appreciated!). Interesting that more rigidly balanced Scapegoat trees couldn’t even complete this test (even when I rewrote tree flattening algorithm to iterative from recursion). My explanation is that it is not very good scenario for Scapegoat tree, as constant deletion triggers re balancing at the root too often (scapegoat tree re balances at the root node on deletion) and it is an expensive operation as it has to flatten and then rebuild the whole tree (not just a scapegoat sub tree as in inserts). However it should do OK when there are not too many deletes in a row. Loosely balanced Scapegoat tree did quite well however. JDK Red-Black tree was fastest and Splay tree was slowest once again.

Delete elements benchmark Score Score error Units AVL tree 88.266 4.713 ms/op JDK TreeSet 67.010 5.138 ms/op Red-Black tree 114.527 12.326 ms/op Scapegoat tree (alpha 0.9) 75.266 1.758 ms/op Splay tree 151.600 8.451 ms/op Treap 97.886 7.516 ms/op

Now lets check searching 100K size trees. Surprisingly JDK Red-Black tree slightly outperformed also very fast AVL tree. Low alpha Scapegoat tree was just slightly slower than AVL tree. Because of splaying overhead on each search operation Splay tree is performing very poorly in such situation.

Search elements benchmark Score Score error Units AVL tree 29.360 1.719 ms/op JDK TreeSet 26.630 4.192 ms/op Red-Black tree 45.811 4.770 ms/op Scapegoat tree (alpha 0.6) 30.151 1.285 ms/op Scapegoat tree (alpha 0.75) 34.001 2.171 ms/op Scapegoat tree (alpha 0.9) 31.732 1.892 ms/op Splay tree 91.072 5.093 ms/op Treap 31.564 1.688 ms/op

This benchmark was specially created for the Splay tree. It is the same as previous search elements benchmark, just now we are constantly searching for only the same 100 elements. Splay tree “is splaying” those to the top after search and that should have given it a superior performance! However probably my benchmark or implementation is flawed in some way and Splay tree under performed!

Search same few elements benchmark Score Score error Units AVL tree 2.301 0.192 ms/op JDK TreeSet 2.667 0.182 ms/op Red-Black tree 15.921 4.166 ms/op Scapegoat tree (alpha 0.6) 1.993 0.192 ms/op Scapegoat tree (alpha 0.75) 2.029 0.110 ms/op Scapegoat tree (alpha 0.9) 1.870 0.196 ms/op Splay tree 19.690 0.761 ms/op Treap 7.887 0.392 ms/op

That is it. As always you can check out all source code from GitHub repo. It is pretty clear my Red-Black tree is not implemented correctly and is inefficient. And probably Splay tree too. It would be great if someone could take a look into it (and other implementations). I hope you liked this little benchmark and it did help you to choose your favorite binary search tree 🙂 Happy coding!

EDIT: check out AVL with RedBlack tree performance comparison here