Suppose a binary search tree with 1000 distinct elements is also a complete binary tree. The tree is stored using the array representation of binary heap trees. Assuming that the array indices start with 0, the $3^{\text{rd}}$ largest element of the tree is stored at index ________.

In a $0$-indexed array representation of a complete binary tree:

• The root is at index $0$.

• The left child of a node at index $i$ is at $2i + 1$.

• The right child of a node at index $i$ is at $2i + 2$.

• The parent of a node at index $i$ is $\lfloor (i-1)/2 \rfloor$.

In a Binary Search Tree (BST), larger elements lie in the right subtree. Following the right child links from the root gives the path to the largest elements:

• Level 0: Index $0$

• Level 1: Index $2(0) + 2 = 2$

• Level 2: Index $2(2) + 2 = 6$

• Level 3: Index $2(6) + 2 = 14$

• Level 4: Index $2(14) + 2 = 30$

• Level 5: Index $2(30) + 2 = 62$

• Level 6: Index $2(62) + 2 = 126$

• Level 7: Index $2(126) + 2 = 254$

• Level 8: Index $2(254) + 2 = 510$

The right child of index $510$ would be at $2(510) + 2 = 1022 > 999$, so index $510$ has no right child. Thus:

• $1^{\text{st}}$ largest element is at index $510$ (the maximum element in the BST).

• Since index $510$ has no left child ($2 \times 510 + 1 = 1021 > 999$), the $2^{\text{nd}}$ largest element is its parent, index $254$.

• The $3^{\text{rd}}$ largest element is the maximum element in the left subtree of index $254$, which is its left child: $2(254) + 1 =$ $509$.

Therefore, the $3^{\text{rd}}$ largest element is stored at index $509$.