How to get the directory tree of files I have read premission on

Two methods come to mind, find and tree. Of the two I think tree fits your question better.

Find
find has the advantage that it is a basic package likely to be found in any linux distro. One disadvantage is that it will not generate a tree view. Nor will greping it for errors be easy. Instead add appropriate -perm tests. The following example will find all directories in / with user, group, and other read permission and send standard error to /dev/null

find / -type d -perm /u=r -perm /g=r -perm /o=r 2>/dev/null

Tree
While tree may not be installed by default it is in the repos. It does create a tree view. If the user running tree does not have read permission a simple grep -v error will remove the directory from the output. However removing the errors will potentially leave extraneous formating. -d List directories only. -p Print the file type and permissions for each file (as per ls -l). grep -v prints lines not matching the pattern, error.

tree -d -p / | grep -v error