Mobile edge computing is a new cloud computing paradigm, which makes use of small-sized edge clouds to provide real-time services to users. These mobile edge-clouds (MECs) are located in close proximity to users, thus enabling users to seamlessly access applications running on MECs. Due to the co-existence of the core (centralized) cloud, users, and one or multiple layers of MECs, an important problem is to decide where (on which computational entity) to place different components of an application. This problem, known as the application or workload placement problem, is notoriously hard, and therefore, heuristic algorithms without performance guarantees are generally employed in common practice, which may unknowingly suffer from poor performance as compared with the optimal solution. In this paper, we address the application placement problem and focus on developing algorithms with provable performance bounds. We model the user application as an application graph and the physical computing system as a physical graph, with resource demands/availabilities annotated on these graphs. We first consider the placement of a linear application graph and propose an algorithm for finding its optimal solution. Using this result, we then generalize the formulation and obtain online approximation algorithms with polynomial-logarithmic (poly-log) competitive ratio for tree application graph placement. We jointly consider node and link assignment, and incorporate multiple types of computational resources at nodes.