How to Search in a Binary Search Tree (BST) – Step by Step Explanation with Examples

mike — Sat, 02 May 2026 23:54:59 +0000

In this video we walk through how to perform searches in a Binary Search Tree. Starting with a perfectly balanced BST, we cover the search process step by step, including finding existing nodes like 73 and determining that numbers like 44 do not exist in the tree.

We discuss time complexity for searches in BSTs – O(log n) on average for balanced trees and O(n) in the worst case when the tree becomes skewed like a linked list due to sorted or bad data. Learn why each comparison eliminates half the remaining data and how height affects performance.

Perfect for students learning data structures and algorithms. If you’re studying BSTs, this clear explanation will help you understand searching, insertion paths, and why self-balancing trees matter.

Thanks for watching!

00:00 Introduction to BST Search
00:31 BST Not Self-Balancing
00:40 Average Case Log Time
00:53 Linear Time in Worst Case
01:13 How BST Search Works
01:18 Search Path Example
02:08 Searching for Existing Node
02:35 Searching for Non-Existent Node
03:19 Tree Size and Height
03:30 Time Complexity O(h)
05:40 Bad Data Example
08:23 Skewed Tree Like Linked List
09:53 Linear Time in Worst Case
11:58 Why Log Time
12:36 Halving the Search Space
14:14 O(log n) Summary
14:28 Conclusion

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hey there let’s perform searches in a binary search tree

okay if you watch my previous videos you know how to define a binary search tree by now

you know some terminology you know how to construct a tree and so now let’s just take

a tree that’s already been constructed and let’s search through it first thing i want to say is

first thing I want to say is this is a binary search tree this is not a self

balancing tree so in the future this tree could get lopsided and messed up

depending on what kind of data we add we can always expect that binary search

trees have a search time or an insert time or removal time of log log base 2

of n on average assuming that the data is random or in the worst possible case

scenario linear time so these could get slower right now you can see this tree

Right now you can see this tree is perfectly balanced.

So if we say that our tree is currently perfectly balanced and it’s always going to be perfectly balanced,

which is a fantasy, but let’s just say that it is,

then we could just say it’s a log tree.

It’s going to take log time to search through.

So how do we perform searches?

Again, if you saw my previous videos, you should know that it’s really easy to find out where a node belongs.

Like for example, if I was going to add the number 14,

I would first look at the root node and I’d say, okay, the root node is occupied.

and I’d say, okay, the root node is occupied.

And so where would 14 belong?

Well, it would belong on the left of 37

because it’s less than 37.

It would belong on the left of 16

because it’s less than 16.

It would belong on the right of 9

because it’s greater than 9.

It would belong on the right of 11

because it’s greater than 11.

And so the 14 node that we just wanted to add,

is that what I just said, 14?

Would end up going there.

So when you’re searching through a binary search tree,

you really just have to go find

have to go find where the node would belong as if you’re performing insert or you could think of it

the other way around like insert is really thinking of a search first but basically we’re just going

to start by saying you know pick a number and we’ll see if it exists so uh let’s pick the number

67 or actually let’s pick the number 73 just to see if it exists in the tree so 73 i’m going to

do like you know seven three with like a question mark we check the root node 73 belongs on the

side we look at the 59 it belongs on the right side and we found our 73 so 73 exists in the tree

and it only took us three examinations only three examinations let’s look for a number that doesn’t

actually exist in the tree so maybe oh my thing is freezing what’s going on okay so maybe uh let’s

The 37 here, 44 belongs on the right of 37.

We look at the 59, it belongs on the left.

We look at the 48, it belongs on the left.

We look at the 43, it belongs on the right.

The 44 would have been there, but it’s not because that right child pointer off the 43

is null.

So we only had to really examine one, two, three, four nodes in order to determine that

our number 44 didn’t exist in the tree.

Notice how we’re not doing a lot of examinations compared to the total size of the tree.

compared to the total size of the tree.

So let me just write this up here.

How many nodes do we have?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.

We got 15 nodes up here.

Someone say N is equal to 15.

And what’s the height of the tree?

Remember, the height of the tree is the number of nodes you must touch

as you make your way down towards the deepest node.

Or if you’ve already found the deepest node,

you take its depth and you just add 1, assuming that depths start at 0.

start at zero. So I’ll just say that the depth of the root node is zero and the 59 and the 16 are

one. And then that third row is two. And that last row is three. And then I just add one to the

deepest node that I can find, which is just going to have a depth of three. All of these on the

bottom row have a depth of three. So the height of this tree is actually four, because we would

so like 37 59 48 43 that’s four touches um or the or the depth of the deepest node plus one okay um

all trees all binary search trees whether they’re actually avl trees or regular binary search trees

or whether they’re balanced or imbalanced they’re always going to have a time complexity to search

through of o of h meaning in the worst case scenario you’ll have to touch h number of nodes

order to find a node by search query or to add a new node or even to delete a new node.

So how does that compare to this tree in reality?

Let me just open up a little calculator real fast.

I just want to show you.

So if we do log base two, because we have a binary search tree, not a trinary search

nodes 15 it tells us that it should take us no more than four examinations to find the node that

we’re looking for if the tree was perfectly balanced and if you if you if you notice that’s

exactly what the height is telling us here if the tree was more imbalanced the height would get

really bad compared to the number of nodes and it would no longer be considered a log tree it would

just be you know somewhere in between linear time and log time um let’s see what else can i show you

let’s do well we did that computation let’s do a search through the tree that

through a tree that’s actually really really bad let’s pretend that we have a

bunch of nodes here and you know I’ve got an old story that I love to tell

about this when like when my grandma was alive she was kind of bitter towards the

end you know I loved her but she used to call the cops on her neighbors for

gossip and at some point in time she got into a feud with her neighbor who was

with her neighbor who was also a bitter old lady.

You know, towards the end, it kind of happens sometimes.

It sucks, but it happened.

And I guess the neighbor didn’t like my grandma’s tree

that she had in her backyard.

So they fought over it, they argued over it,

and then one day my grandma said

she was looking out the backyard window

to yell at ducks who were jumping into her pool

because she just hated when the ducks went into her pool.

She was waiting for them to show up.

She wasn’t even looking at ducks.

even looking at ducks she was just waiting and she saw the other little old lady reach her arm over

the fence and spray my grandma’s tree with some sort of poison then my grandma’s tree died and uh

i guess you know my grandma never never got over it i always thought it was like a funny story it

was a little sad but um she poisoned my grandma’s tree so we can kind of do the same thing with the

binary search tree we can give bad data let me show you what happens here if i use this data set

search tree. So we’re going to add this node right here. It’s going to be the 14.

And then when we add the 19 node, I’m just going to duplicate it. The 19 is supposed to be on the

right side of 14 because it’s greater than 14. So I’m just going to like, you know, draw a 19 right

here and a little connecting line. Okay. Then when we add the 24, I’m going to duplicate this node

the 24 go belongs on the right side of the 14 and also belongs on the right side of the 19

because it’s greater than 19 so again we have a node that shows up on the right side let me

remember to actually fix the 19 so it looks like 24. then when we add the uh 29 it’s gonna end up

being the same thing right the 29 is gonna end up showing on the right the right the right it’s

If we have bad data, we can end up with a tree that kind of looks like a straight line.

And if you are watching this video from the future,

after I’ve posted all of my data structures videos that I have,

you might notice that this looks like another data structure.

If you’re watching this right away, I haven’t posted those videos just yet.

But just tilt your head to the right a little bit

and tell me if that looks like something that you recognize.

It looks like a linked list.

looks like a linked list just like a straight line or even i guess you could kind of say maybe an array

um just like a straight line of data there’s no uh you know there’s no splitting of the data

so we have the 29 and then the 35 and then uh maybe i’ll just do one more node only and i’ll

get rid of that 54 because that’s too many nodes so what happens is if you have you know bad data

tree doesn’t actually search or add insert or remove or anything like that in log time anymore

instead it operates in linear time just like a linked list would meaning we would have to scan

through every single node in the entire tree to be sure that some value did not exist well

not exactly but in the worst case scenario we’d have to scan through the whole entire tree

for example let’s just look at the the number of nodes real fast we’ll say the

number of nodes here is one two three four oh I put two 24s in there oh dearie

let me get that says 14 19 24 29 and then I guess I got to make that a 35 oh

okay and then there’s a 49 I guess I did add every single node okay so we added

so there’s like seven uh items in the tree so i’m going to say n is equal to seven the height also

is seven though so the height is seven notice how if we say that it’s always true that a binary

search tree has a time complexity in the worst case of o of h meaning you’d have to go down

in the worst case the number of levels that is equal to the whole entire height of the entire

tree that’s also the number of nodes in the tree which means uh if i wanted to add let’s say the

I don’t know, if you wanted to add the number 60,

60 would end up going down here.

And in order to find where 60 would go,

we’d have to look at the 14, that’s 1, 19, 24.

We’d have to look at all these nodes.

That is seven nodes.

We’d have to examine seven nodes

just to figure out where the 60 would go.

Or if we were searching for 60,

then we’d have to go down that far

to find out that 60 didn’t exist.

So this is nowhere near a log tree.

This is a linear tree, a linear time tree.

year time tree and just to drive the point further if we did log base two of the number of nodes in

this case seven notice how it’s telling us that we should if it was a log tree if it was perfectly

balanced or a self-balancing tree we should have to touch no more than three nodes in order to find

what we’re looking for to search through the tree or to insert a new node or even to delete a node

but that’s not the case here because the tree is really really slow compared to a perfectly balanced

slow compared to a perfectly balanced tree.

So we call this a linear tree.

It sucks.

You want to try to avoid that.

You don’t want your data to be poisoned.

On average, your trees won’t really look like this if you just have totally

random data, but if you think there’s a chance that your data might be skewed

in some way, it’s probably a good idea to upgrade to a self-balancing tree,

which we’re going to talk about later.

Okay.

What else can I tell you real fast?

So we talked about searching.

why are these trees considered log time? I mean, they’re really, really fast, but why?

Let’s just duplicate this slide here and I’ll move it down one level. I just want to kind of

show you why this ends up being a log time tree. Okay. So why, why is this a log time tree?

Suppose for the sake of argument, we’re searching for the number 52. So if we’re searching for the

node the root node and decide where would the 52 belong let me actually write that down so i don’t

forget 52 question mark where would the 52 belong it would belong on the right side of the 37 because

it’s greater than 37 right which means it’s actually impossible that the 52 would be in any

of these other nodes on the left side there’s no need to check them we’re not going to scan the

whole tree we’re just going to go left or right and ignore the whole other side of the tree

we just did is with one examination we eliminated half of the remaining data set or i guess of the

initial data set if we’re looking at the root node so then the next thing that we do is we just kind

of say okay it belongs on the right side so we look at the 59 node and we do the same thing we

say where would 52 be would it be on the left side or the right side of the 59 node it would be on

the left side which means we’re going to eliminate from consideration the entire right subtree of the

So again, we’re eliminating half of the remaining data set.

Then we kind of go down here and we look at the 48.

We do the same thing.

52 belongs on the right side.

So we eliminate half of the remaining data set, you know, and then eventually we do find

the 52 and then that’s it.

So if we’re eliminating half of the data set with every single, just one examination at

a time, that’s why it ends up being log base two.

getting twice as close to your destination at every step you took if you’re like in a car like

running maybe you’re on the moon or something then it would be a log time you know uh progress

towards your goal or eliminating half of the data set at every single step sometimes people see

for loops on exams and stuff when we talk about this um like for i equals zero uh i is less than

it would be like i times equals two.

So i’s jumping twice as far each time.

That’s the sort of thing that makes it log.

So I’m just going to write down log here real fast

to drive the point home a little bit more.

O of log base two of n.

Yeah, so I guess that’s all I wanted to tell you in this video.

Hopefully you feel like an expert in searching now.

If you want to see more searching videos,

binary tree search Archives - NeuralLantern.com

How to Search in a Binary Search Tree (BST) – Step by Step Explanation with Examples