Still learning the ropes here and probably have some very ill formed code - so, any help/advice is greatly appreciated.

Here is the scenario of what I am trying to accomplish:

We have two custom objects (related via lookup field on Essay object):  Applications & Essays.  I have created a custom field on the Applications object called XEssays_On_File.  When an Essay record is added/updated, I would like the XEssays_On_File field to get updated with the number of related Essay Records. (Can't use rollup fields as this is not a master-detail relationship)

I have drafted the following code - which works for single situations - but fails mercilessly in bulk update:

trigger CountRelatedEssays on TargetX_SRMb__Essay__c (after insert, after update) {

        TargetX_SRMb__Essay__c [] ess = Trigger.new;
        String appid = null;
        appid = ess[0].TargetX_SRMb__Application__c;
       
        Integer i = [select count() from TargetX_SRMb__Essay__c where TargetX_SRMb__Application__c =     :appid];
       
        TargetX_SRMb__Application__c [] app =[select id, XEssays_On_File__c from TargetX_SRMb__Application__c where id = :appid];

        app[0].XEssays_On_File__c = i;   
       
        update app[0];
       

}

I would also like this to fire and update on Delete as well.

Thanks in advance for any help you can provide!