On the Mac, both the yEd installer and yEd itself, when double-clicked in the Finder, need a Java 6 JRE to run.
(If there is a JRE 7 installed on your Mac in Library/Internet Plug-Ins, then this JRE is actually only intended for use by your browsers.)
If you don't want to install a Java 6 JRE, you can run yEd via Web Start from our website. Alternatively, you can download the yEd Zip file, which contains "yEd.jar". If you double-click this .jar file, it should trigger the JRE 7.
That said, we still believe that Java 7 on the Mac is not ready for prime time yet. Currently, for best experience on the Mac, we recommend to run yEd with JRE 6.