You don't need root access to install Tomcat in Linux, and
you should NOT run Tomcat as root, for security reasons. As long as you don't reassign any of Tomcat's ports down below the magic line (port 4096), you don't need to run as root.
The Tomcat zip file has no absolute filepaths in it, so where you install it (and even what OS you install it under!) are of no importance.
Of course, from a sysadmin's point of view, there are some "preferred" places and some of them may require temporary root permission to unzip into. Historically, /usr/local is popular, but Tomcat has some writable directories in it, and according to the LSB, /usr is supposed to be read-only these days. You can get around this by strategically configuring CATALINA_HOME and CATALINA_BASE. Or you can use one of the alternative choices which are less regulated. /opt used to be good, but these days, /srv is probably better.
For personal use, just plunk Tomcat somewhere under your home directory.