Minimum Spanning Tree (MST) and Breadth-First Search (BFS) tree constructions are classical problems in distributed computing, traditionally studied in the message-passing model, where static nodes communicate via messages. This paper investigates MST and BFS tree construction in an agent-based network, where mobile agents explore a graph and compute. Each node hosts one agent, and communication occurs when agents meet at a node. We consider $n$ agents initially dispersed (one per node) in an anonymous, arbitrary $n$-node, $m$-edge graph $G$. The goal is to construct the BFS and MST trees from this configuration such that each tree edge is known to at least one of its endpoints, while minimizing time and memory per agent. We work in a synchronous model and assume agents have no prior knowledge of any graph parameters such as $n$, $m$, $D$, $Δ$ (graph diameter and maximum degree). Prior work solves BFS in $O(DΔ)$ rounds with $O(\log n)$ bits per agent, assuming the root is known. We give a deterministic algorithm that constructs the BFS tree in $O(\min(DΔ, m\log n) + n\log n + Δ\log^2 n)$ rounds using $O(\log n)$ bits per agent without root knowledge. To determine the root, we solve leader election and MST construction. We elect a leader and construct the MST in $O(n\log n + Δ\log^2 n)$ rounds, with $O(\log n)$ bits per agent. Prior MST algorithms require $O(m + n\log n)$ rounds and $\max(Δ, \log n) \log n$ bits. Our results significantly improve memory efficiency and time, achieving nearly linear-time leader election and MST. Agents are assumed to know $λ$, the maximum identifier, bounded by a polynomial in $n$.