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I'm studying Binary Search Trees (BST) and I would like to verify that my understanding of BSTs is correct.

For example, let S = [17, -10, 7, 19, 21, 23, -13, 31, 59].

Binary Search Tree for S, with S1 as root:

enter image description here

How many comparisons is needed to search this tree for the presence of each of the elements in I = {-13, 31}?

Search for -13:

Since -13 is less than 17: Go to left Child Is -13 < 17?

Comparisons = 1

Since -13 is less than -10: Go to left Child

Comparisons = 2

Item (13) found with two comparisons.

Search for 31:

Since 31 is greater than 17: Go to right Child

Comparisons = 1

Since 31 is greater than 21: Go to right Child

Comparisons = 2

Sincer 31 is greater than 21: Go to right Child

Comparisons = 3

Item (31) found with three comparisons.

Total number of comparisons to find the presence of each item in I: 5.

Is my understanding correct?

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Yes... You can also practice with an implementation of your own. I have created such an implementation and a good test is to compile this with different optimization levels in gcc:

#include <stdio.h>
#include <stdlib.h>
#include <time.h>

typedef struct node {
    int node_id;
    int data;
    char *name;
    struct node *left;
    struct node *right;
} node;

node *newNode(int data, char *name, int node_id) {
    node *new_node = malloc(sizeof(node));
    new_node->data = data;
    new_node->name = name;
    new_node->node_id = node_id;
    new_node->right = new_node->left = NULL;
    return new_node;
}

node *insert_node(node *root, int data, int node_id, char *name) {
    if (root == NULL)
        return newNode(data, name, node_id);
    else {
        node *cur;
        if (node_id < root->node_id) {
            cur = insert_node(root->left, data, node_id, name);
            root->left = cur;
        } else if (node_id > root->node_id) {
            cur = insert_node(root->right, data, node_id, name);
            root->right = cur;
        }
    }
    return root;
}

node *find_node_data(node *root, int node_id) {
    if (root == NULL)
        return NULL;
    else if (root->node_id == node_id)
        return root;
    else if (root->node_id > node_id)
        return find_node_data(root->left, node_id);
    else
        return find_node_data(root->right, node_id);
}

void print(node *np) {
    if (np) {
        print(np->left);
        printf("(%d, %d, %s)", np->node_id, np->data, np->name);
        print(np->right);
    }
}

int main() {
    int T = 1000; //test case 1000 nodes
    int data, r2, r;
    int node_id = 0;
    //printf("Input number of nodes:");
    //scanf("%d", &T);
    node *root = NULL;
    srand(time(NULL));
    while (T-- > 0) {

        //scanf("%d %d", &data, &node_id);
        r = (2 + T) * rand() % 100; //node id
        r2 = (2 + T) * (rand() % 100); // data
        printf("Input data. %d:\n", r2);
        printf("node id. %d:\n", r);
        root = insert_node(root, r2, r, "foobar");
    }
    print(root);
    // printf("\n");
    // printf("Find data at node:");
    // scanf("%d", &T);
    print(find_node_data(NULL, node_id));
    T = r;
    printf("node data %d %s", find_node_data(root, r)->data, find_node_data(root, r)->name);
    root = NULL;
    // find_node_data(root, T)->data;
    print(root);
    return 0;
}
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