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So Leo introduced binary search trees in the previous video, and

what we'd like to do now is really focus in on the performance of algorithms that

use binary search trees,

because with all of these data structures that we're talking about, one of the big

motivations for introducing a data structure is to get better performance.

So let's see if we do that with binary search trees.

So in particular, by the end of this video,

you'll be able to explain the running time performance of looking for

a word in a binary search tree and comparing this performance with other data

structures that we have where we might want to search for words.

So, in particular, for linked lists.

All right, so let's think back to binary search trees and, in particular,

in the programming assignment that you were working through we'll be using

binary search trees to store dictionaries.

So we are storing lists of words and we like to be able to lookup and search for

words in this data structure.

So we're gonna do just a small example with these words, and

let's think about what binary search trees we might

get if we start building a tree using these words.

So it turns out that it's not just a single binary search tree that

might result.

We could get any of these three or others potentially.

Depending on the order in which we insert words into the tree.

And so you can follow along and each one of these trees and confirm that,

in fact, when you look at a node that's to the left of a node,

and a node that's on its right, each time you're going to get the one

that's on the left is alphabetically before the one that's on the right.

Whether it be in the green,

binary search tree where we all have just a single path going down or

in one of the bushier trees that you see on the right hand side as well.

All right, so we have these three different binary search trees.

All try to capture that same set of words that form our dictionary.

And that's going to be interesting when we think about performance because we

have to think about each of these different possible

binary search trees that we might be working with.

So let's think about the isWord method for looking up a word

in the binary search tree, so looking up a word in the dictionary.

You can think about an application where we were just building the tree and

we're trying to add new words to it.

And so every time we're looking up to see if it's already in there and

then perhaps adding to it.

Or you can think about searching through a big collection of data and

trying to find some particular piece that we care about.

In either situation what we want to do is we want to start at the root

of the tree, and

we want to compare the word that we care about to the value at the current node.

Now, if at the current node we actually don't have a word, if the current is null,

then that means that we haven't found our word that we care about.

It's not in the tree and so we return false.

On the other hand, what we want to do is we wanna compare

the word that we care about to the current word, and if it actually agrees with it.

Then we wanna return true.

And so if we found the word, then we're happy, and we can return true.

On the other hand, we have two possibilities for what happens if our node

isn't null and if the current word doesn't agree with the word that we want to find.

Either the word that we want to find is alphabetically before the current word, or

it's alphabetically after the current word.

And if it's alphabetically before the current word,

then we only need to look at the sub tree to the left of the current word.

If the word that we care about is alphabetically greater than or

after the word that we're looking at right now,

then we need to look at the right sub tree of the binary search tree.

All right, so, let's think about what that might look like and

remember when we're analyzing performance we want to think about best case and

worst case performance, you want to think about average case.

And with each situation it's useful to work through some examples and

see what does it mean to get lucky when we're running this method on

particular inputs, and what does it mean to get unlucky?

And also we need to think about what size of input do we,

what does it mean to talk about the size of the input?

What input do we have to this method?

And when you think about this isWord method,

what we're doing is searching through a collection.

And so the relevant input size, is the size of the dictionary.

The number of words we're looking through.

It's not so much the length of the current word.

It's not so much how many words we're looking for but,

it's the size of the dictionary, the number of words

that we're storing that we're looking through in order to find the word to find.

Okay, so our input size, or n, is the number of words in our dictionary.

And what we'd like to think about is how to relate the performance of this method

is word, to that size.

How does our performance change as the number of words in the dictionary changes?

So let's, for example,

look at what happens when we run isWord on the word east.

And what we have to do using our algorithm is look first at the root

of the binary search tree.

And oh yay, we're lucky, we hit our desired word, east, right away.

And so that means that for this particular run of the algorithm, we've got just

a single comparison that we make between the argument and a value of the tree.

And so, well that comparison might take a few operations, but

if we are lucky in the very first probe of the binary search tree, the very first

comparison we make, that's going to take a constant number of operations.

It's not going to depend on how big the tree was if we already

were successful when we looked at the root.

So in the best case, the case where we find our word right away,

the performance of this algorithm is big O(1), constant time.

But, we're not always lucky.

So, let's look at another case where the word that we're looking for

isn't at the root.

So let's think about what happens when we look for the word, a.

So we start at the root.

And we compare east and a, and we notice that a happens alphabetically before east.

So we have to look at that left sub-tree of the binary search tree.

Now we're looking at comparing a with at.

All right, a is still alphabetically at at.

It's a prefix of at so it comes earlier.

So we look to the left.

And now we're comparing a with am.

All right, now again a is alphabetically before am, and so

we have to look to the left, but we didn't find anything there.

The children of am are null.

And so what we notice is we've fallen off the tree.

A does not occur in our dictionary.

And so we can return an answer that says It is not a word in our dictionary so

false.

Now that's okay, it's okay if we don't always find the word, but

what we are curious about here is the performance.

How long did it take us to come to this conclusion?

And notice that we compared here our words to find a with three out of a seven words

in the dictionary and so we're working through examples, but we really care about

the general case so can we extrapolate what it means to think about three out of

the seven words to a general case as the dictionary gets really big.

Well, that seems hard.

So, let's think about maybe some more examples that will give us

insight into the general case.

And in particular we have to remember that our binary search tree

might be organized a little bit differently.

So let's think about the exact same input a but with a different

characterization of the binary search tree so a different organization of it.

And so if we had that linear version of the binary search tree what happens,

how many comparisons we need to do as we look for the word a amongst the seven

words in the dictionary that we populated the search tree with.

So again, we compare first to the root and we see that a comes before e, so

we go to the left.

We compare with the next word.

A is still before e and so we go to the left.

We compared a with e here, and we see that we still have to go left.

We compare a with ate, we still have to go left.

We compare a with at, we still have to go left.

We compare a with m, and we notice that now we fall off the tree, and

we see that a is not in there.

But we have to compare with every single one of the words in our dictionary

in order to come to this conclusion.

So notice that this was even worse than before

in terms of the number of comparisons that we had to do.

And so if we're doing worst case analysis,

then this is the kind of tree that we need to think about.

So, not only did we have to think about the number of elements in the dictionary,

so here, n was seven, we also had to think about how the tree was organized,

and what organization would lead us to the worst case behavior,

the worst possible behavior.

And, so, we see that in this linear tree, when we're looking for a word

that's not on the tree, in the worst case we have to make o of n comparisons.

We had to compare our word with every single one of the words in the dictionary.

Okay, so, that leads us to this analysis of how can we avoid the worst case?

So in the worse case we had this linear tree.

And we had to go all the way to the bottom to the first time that we fell

off the tree.

And then we're only then we'll be able to return false.

But maybe we can think of a modification of our data structure,

that will help us avoid the situation.

And so what we'd like to do is think about how the path that we have

to travel until we fall off the tree can be controlled.

Or if there is a property of the tree that we can impose that will make sure that we

don't have to go too far.