Graph similarity is an important operation with many applications. In this paper we are interested in graph edit similarity computation. Due to the hardness of the problem, it is too hard to exactly compare large graphs, and fast approximation approaches with high quality become very interesting. In this paper we introduce a novel upper bound computation framework for the graph edit distance. The basic idea of this approach is to picture the comparing graphs into hierarchical structures. This view facilitates easy comparison and graph mapping construction. Specifically, a hierarchical view based on a breadth first search tree with its backward edges is used. A novel tree traversing and matching method is developed to build a graph mapping. The idea of spare trees is introduced to minimize the number of insertions and/or deletions incurred by the method and a lookahead strategy is used to enhance the vertex matching process. An interesting feature of the method is that it combines vertex map construction with edit counting in an easy and straightforward manner. This framework also allows to compare graphs from different hierarchical views to improve the upper bound. Experiments show that tighter upper bounds are always delivered by this new framework at a very good response time.