1. A computer program product for maintaining data integrity in a computer data storage system by backing up and restoring one or more clone file trees in the computer data storage system, the computer program product comprising: one or more non-transitory computer readable storage media and program instructions stored in the one or more non-transitory computer readable storage media, the program instructions that are executed by at least one or more computer processors comprising: program instructions to generate a clone management table that includes at least one clone file parent and at least one clone file child; program instructions to record, in the clone management table, a clone management information about each clone file in one or more clone file trees in a file system of a computing device, wherein (i) the clone file tree includes metadata specifying a sequence of files with parent-child relationships in which a given clone file child is generated from a given clone file parent that immediately preceded the given clone file child in the sequence (ii) the given clone file parent is modified to be a read-only file when a given clone child file is created and (iii) the given clone file child is a read and write file; program instructions to, in response to receiving a backup request, update the clone management information and store one or more data blocks that are in one or more clone file trees; program instructions to, in response to receiving a request to restore a clone file, analyze, the clone management information about the clone file and restore one or more data blocks in the clone file to the file system; program instructions to, in response to receiving a request to restore a clone file parent, analyze the clone management information about the clone file parent and restore one or more data blocks in the clone file parent to the file system, wherein a given clone parent the contains all data blocks that were originally in a file when a clone of the given clone parent file was created; program instructions to, in response to receiving a request to restore a clone file tree, (i) analyze the clone management information about the clone file tree and (ii) restore one or more data blocks in the clone file tree to the file system of the computing device; and program instructions to, in response to receiving a request to check a validity of the clone management table, analyze the clone management information in the clone management table for coherence, and generate a list of an invalid information that is in the clone management table, wherein the updating the clone management information in the clone management table enables the backup and the restoration of at least a part of the clone file tree precludes one or more data redundancies and restores the clone file tree to a form that the clone file tree had when the clone file tree was backed up.