1. A computer-implemented method, comprising: identifying, on a computing device, a parent component in a virtual-machine environment, wherein the parent component includes a host cluster of two or more hosts, a host, or a set of two or more virtual machines; identifying a set of two or more child components having at least one of the following relationships to the parent component: when the parent component includes a host cluster of two or more hosts, each child component in the set of two or more child components includes a host in the host cluster, when the parent component includes a host, each child component in the set of two or more child components includes a virtual machine running on the host, and when the parent component includes a set of two or more virtual machines, each child component in the set of two or more child components includes a virtual machine in the set of two or more virtual machines; determining, on a computing device, a performance metric for each child component in the set of two or more child components, wherein the performance metric characterizes a task-processing time, a count or percentage of task completions, an intake number, a resource power state, a processor usage, a processor speed, a memory usage, a network usage, an occurrence of memory swapping or memory ballooning, an occurrence of an error, a task migration variable, a processing-power allocation, or a memory allocation; determining a child-component performance state for each child component in the set of two or more child components based on the performance metric for the child component and a child-component state criterion, wherein the child-component state criterion maps a first range of values for a performance metric to a first child component performance state and a second range of values for the performance metric to a second child-component performance state; determining a parent state for the parent component based on the child-component performance state for each child component in the set of two or more child components and a parent-component state criterion, wherein the parent-component state criterion includes a threshold percentage or number of child components that have a specified state; displaying a visual representation of the virtual-machine environment as a hierarchical tree, wherein the visual representation includes a representation of the parent component and an indication of the parent state of the parent component, but does not include a representation of the two or more child components; when a user selects the parent component through a user interface, displaying a representation of the set of two or more child components, and an indication of the child-component performance state for each child component in the set of two or more child components; and when the user selects one of the two or more child components through the user interface, displaying a representation of a set of two or more lower-level child components of the selected child component, and an indication of a performance state for each displayed lower-level child component of the selected child component.